Title: Theory / Important / Math / Al gebra / Abstract Algebra - Dummit - Foote+14{ / 1 Group
Tags: fromMath mathToDef hasSeenConjDef mathToSee automaticImplies examplesSwitchExample1 defToAlgebra closed setMultiplication exampleToQuestion TypToQuestion mid1516974248815 récupéré toAlgebra toUnary Copy counterExample questionExample setPresentation toBinaryRelation todo typToBinary

Question: \(\star:C\times C\to C\) such that
Answer: \(a\star (b\star c)=(a\star b)\star c\).
Page: 016
Chapter: 1
Variables: \(C\) a set
Kind: Def
Name: associative operation

Name: A comutative operation on \(C\)
Definition: A function \(\star:C\times C\to C\) such that \(a\star b=b\star a\).
Page: 016
Section: 1
Variables: \(C\) a set,
Kind: Def

Name: A group
Definition: A set \(G\) with an associative operation \(\star\) which admits an identity element and such that all elements admits an inverse
Page: 016
Section: 1
Kind: Def

Name: An identity of \(\star\)
Definition: \(a\star i=i\star a=a\) for all \(a\).
Page: 017, 20
Chapter: 1
Variables: \(C\) a set,
\(\star:C\times C\to C\).
Index: ,1
Kind: , ex
Definition2: the order of \(i\) is 1
Conjdef: An element \(i\) such that, equivalently

Question: \((a\star b)^{-1}=\)
Answer: \(a^{-1}\star b^{-1} \)
Page: 018
Chapter: 1
Variables: \((G,\star)\) a group and \(a,b\in G\).
Kind: Prop

Question: \((a^{-1})^{-1}=\)
Answer: \(a\)
Page: 018
Chapter: 1
Variables: \((G,\star)\) a group and \(a\in G\).
Kind: Prop

Question: In a group, the equation \(ax=b\) and \(ya=b\) have
Answer: unique solutions for \(x,y\in G\)
Page: 020
Chapter: 1
Variables: \((G,\star)\) a group and \(a,b\in G\).
Index: 2
Kind: Prop

Question: \(au=av\) iff
Answer: \(u=v\)
Page: 020
Chapter: 1
Variables: \((G,\star)\) a group and \(a,u,v\in G\).
Index: 2
Kind: Prop

Name: The order of \(g\)
Definition: the least \(n\) such that \(g^n=1\)
Page: 020
Chapter: 1
Variables: \(G\) a group and \(g\in G\).
Kind: Def
Definition2: the cardinal of the set generated by \(g\).

Name: The multiplication table of \(G\)
Definition: A \(|G|\times |G|\)-matrix with the elements as entry and their product as value.
Page: 021
Chapter: 1
Variables: \(G\) be a group.
Kind: Def
Name2: The group table of \(G\)

Name: The dihedral group of order \(2n\)
Definition: The set of symmetries of a regular \(n\)-gon
Page: 023
Chapter: 1
Variables: \(n>2\)
Notation: \(D_{2n}\)
Definition2: \(\langle s,r\mid r^n=s^2=1,rs=sr^{-1}\rangle\).

Q: [...].
Its order is [...].
A: the rotation by \(2\pi/n\); \(n\)
Page: 024
Chapter: 1
Section: 2
Variables: \(n\in\mathbb N\)
Notation: \(r\) in \(D_{2n}\)

Definition: The reflexion fixing \(1\).
Page: 025
Chapter: 1
Section: 2
Variables: \(n\in\mathbb N\)
Context: \(D_{2n}\),
Notation: \(s\)

Q: In a dihedral group \(s[...]r^i\)
A: {\ne}
Page: 025
Chapter: 1
Section: 2
Variables: \(n\in\mathbb N\)
Index: (3)

Q: In \(D_{2n}\), \(r^j[...]r^i\) for \(0\le i<j<n\)
A: {\ne}
Page: 025
Chapter: 1
Section: 2
Variables: \(n\in\mathbb N\)
Index: (1)

Smaller0: \(sr^{-i}\)
Greater0: \(r^{i}s\)
Page: 025
Chapter: 1
Section: 2
Variables: \(n\in\mathbb N\)
Index: (4)
Smaller to greater: =
Hide middle: yes
Context: \(D_{2n}\)

Question: The elements of \(D_{2n}\) are
Answer: \(\{r^i,sr^i\mid 0\le i<n\}\) .
Page: 025
Chapter: 1
Section: 2
Variables: \(n\in\mathbb N\)
Index: (4)

Name: Set generating \(D_{2n}\)
Page: 026
Chapter: 1
Section: 2
Variables: \(n\in\mathbb N\)
Example: \(\{r,s\}\)

Name: The symmetric group of degree \(n\)
Definition: \(S_{\{1,\dots,n\} }\).
Page: 029
Chapter: 1
Section: 3
Variables: \(n\in\mathbb N\)
Notation: \(S_n\)

Name: A cycle
Definition: \(a_i\mapsto a_{i+1\mod m}\).
Page: 029
Chapter: 1
Section: 3
Variables: \(S\) a set
Notation: \((a_1\dots a_m)\)
Assuming: \(a_1,\dots,a_m\in S\) some distinct elements.

Name: An \(n\)-cycle
Definition: A cycle of length \(n\).
Page: 029
Chapter: 1
Section: 3

Name: Disjoint cycles
Definition: Two cycles which have no numbers in common.
Page: 029
Chapter: 1
Section: 3

Name: The length of \(\tau\)
Definition: \(n\)
Page: 029
Chapter: 1
Section: 3
Variables: \(\tau=(a_1,\dots,a_n)\) a cycle

Question: \(\sigma^{-1}\) is obtained by
Answer: rewritting the elements in the reverse order.
Page: 029
Chapter: 1
Section: 3
Variables: \(\sigma\) a cycle

Question: Disjoint cycles
Answer: commute.
Page: 029
Chapter: 1
Section: 3

Question: \((a_i\dots a_m a_1\dots a_{m-1})=\)
Answer: \((a_1\dots a_m)\)
Page: 029
Chapter: 1
Section: 3
Variables: \(S\) a set and \(a_1,\dots,a_m\in S\) distinct.

Name: \(\star\) is distributive over \(\cdot\)
Definition: \(a\cdot(b\star c)=a\cdot b\star a\cdot c\).
Page: 034
Chapter: 1
Section: 4
Variables: \(G\) a set, \(\star, \cdot:G\times G\to G\).

Definition: \(\mathbb F\setminus\{0\}\)
Page: 034
Chapter: 1
Section: 4
Variables: \((\mathbb F,+,\times)\) a field.
Notation: \(\mathbb F^\times\)
Kind: Def

Name: The general linear group of degree \(n\)
Definition: The set of \(n\times n\) matrices with entries in \(F\) and non nul determinant.
Page: 034
Chapter: 1
Section: 4
Variables: \((\mathbb F,+,\times)\) a field and \(n\in\mathbb N\).
Notation: \(GL_{n}(F)\)

Question: The cardinal of \(GL_n(F)\) is
Answer: \(\Pi_{i=0}^{n-1} (q^n-q^i)\).
Page: 035
Chapter: 1
Section: 4
Variables: \(F\) a field of finite cardinal \(q\) and \(n\in\mathbb N\).

Page: 036, 845
Chapter: 1, 18
Section: 5, 1
Name: The quaternion group
Set: \(\{1,-1,i,-i,j,-j,k,-k\}\)
Notation: \(Q_8\)
*: \(ii=jj=kk=-1\), \(ij=k\),\(jk=i\), \(ki=j\), \(ji=k\),\(kj=i\) and \(ik=j\).
1: 1
÷: \(1\mapsto 1\), \(-1\to-1\) and \(x\mapsto-x\) otherwise
Commutative: No
Relation: \(i^4=j^4=1\)
Generating family: \(\{i,j\}\)
Relation2: \(i^2=j^2\)
Relation3: \(i^{-1}ji=j^{-1}\)

Name: An homomorphism
Definition: \(\phi:G\to H\) such that \(\phi(x\star y)=\phi(x)\square \phi(y)\).
Page: 036
Chapter: 1
Section: 6
Variables: \((G,\star)\) and \((H,\square)\) be groups and \(\phi:G\to H\).
Kind: Def

Name: An isomorphism of groups
Definition: A bijective homomorphism of groups.
Page: 037
Chapter: 1
Section: 6
Kind: Def

Question: A group of order 6 is
Answer: \(S_3\) or \(\mathbb Z/6\mathbb Z\).
Page: 038
Chapter: 1
Section: 6

Name: A left group action of \(G\) on \(A\)
Definition: A map \(G\times A\to A\) such that \(g(g'a)=(gg')a\) and \(1a=a\).
Page: 041
Chapter: 1
Section: 7
Variables: \(G\) be a group
\(A\) a set.

Definition: [$]\defFun{G}{G}{a}{ga}[/$]
Page: 042
Chapter: 1
Section: 7
Variables: \(G\) be a group,
\(g\in G\)
\(A\) a set on which \(G\) acts.
Notation: \(\sigma_g\)
Typ: A permutation

Name: The permutation representation associated to \(\cdot\)
Definition: The map [$]\defFun{G}{S_A}{g}{\sigma_g}[/$].
Page: 042
Chapter: 1
Section: 7
Variables: \(G\) be a group
\(A\) a set
\(\cdot:G\times A\to A\) a left-action
Typ: Homomorphism

Name: The trivial action of \(G\) on \(A\)
Definition: The action [$]\defFun{G}{\mathcal S_A}{g}{\defFun{A}{A}{a}{a}}[/$]
Page: 042
Chapter: 1
Section: 7
Variables: \(G\) a group
\(A\) a set.

Name: A faithful group action
Definition: A group action such that \(\sigma_g\ne\sigma_{g'}\) for \(g\ne g'\in G\).
Page: 043
Chapter: 1
Section: 7
Variables: \(G\) a group and \(A\) a set.
Kind: Ex

Name: The kernel of the group action \(\cdot:G\times A\to A\)
Definition: the set of \(g\) such that \(\sigma_g\) is the identity
Page: 043
Chapter: 1
Section: 7
Variables: \(G\) a group
\(A\) a set
\(\cdot:G\times A\to A\) a group action.
Kind: Ex

Page: 046
Definition: The set of pairs \((H,G)\) of groups, with \(H\subseteq G\).
Chapter: 2
Section: 1
Notation: \(\le\)
Name: The subgroup relation
Kind: Def
Context: group
Transitive: Transitive

Name: The trivial subgroup of \(G\)
Definition: \(\{1\}\) as a subgroup of \(G\).
Page: 047
Chapter: 2
Section: 1
Variables: \(G\) a group.
Kind: Ex

Name: The normalizer of \(A\) in \(G\)
Definition: \(\{g\in G\mid gAg^{-1}=A\}\) .
Page: 050, 88
Chapter: 2, 3
Section: 2, 1
Variables: \(G\) a group
\(A\subseteq G\).
Notation: \(N_G(A)\)
Index: , 31
Kind: Def, Ex

Q: \([...][...][...]\).
A: C_G(A); \le; N_G(A)
Page: 050
Chapter: 2
Variables: \(G\) a group and \(A\subseteq G\).

Name: An Abelian group
Definition: \(Z(G)=G\)
Page: 22, 050, 339, 171, 40, 89, 134
Chapter: 1, 2, 10, 5, 1, 3
Section: 1, 2,1, 4, 6, 1
Index: 10, 1, v, 17-18, 25, 1
Kind: Ex, Ex, Ex, Ex, Ex, Ex, Ex
Definition2: the inner automorphisms of \(G\) are 1.
Conjdef: A group \(G\) such that, equivalently
Definition3: \(G\) is a \(\mathbb Z\)-module
Name2: A commutative group
Extra: Linear algebra, Kenneth Kunze, Ch 1.1, P 1,2
Abstract Algebra, Summit: Ch 1.1 p22
Ch 2.2, p 50,
Ch 10, 1, p 339
Ch 5. 4, p 171
Ch 1.6, p 40
Ch 3.1 p 89P 134 ex 1
Definition4: \(G'=1\)
Definition5: the group table is a symmetric matrix
Definition6: \(g\mapsto g^{-1}\) is a morphism
Definition7: \(G/Z(G)\) is cyclic
Definition8: \(g\mapsto g^{2}\) is a morphism

Question: If \(G\) is abelian, \(C_G(A)=\)
Answer: \(G\)
Page: 050
Chapter: 2
Variables: \(G\) a group and \(A\subseteq G\).
Index: 1
Kind: Ex

Question: \(C_{D_{8} }(\langle r\rangle)=\)
Answer: \(\langle r\rangle\)
Page: 050
Chapter: 2
Kind: Ex

Question: \(N_{D_{8} }(A)=\)
Answer: \(D_8\)
Page: 050
Chapter: 2
Variables: \(A\) be the subgroup of rotation of \(D_8\).
Index: 3
Kind: Ex

Question: \(Z(D_{8})=\)
Answer: \(\{1,r^2\} \)
Page: 050
Chapter: 2
Index: 4
Kind: Ex

Question: \(C_{S_3}(\{1,(12)\})=\)
Answer: \(\{1,(12)\} \)
Page: 051
Chapter: 2
Index: 5
Kind: Ex

Question: \(N_{S_3}(\{1,(12)\})=\)
Answer: \(\{1,(12)\} \)
Page: 051
Chapter: 2
Index: 5
Kind: Ex

Name: The stabilizer of \(S\)
Definition: \(\{g\in G\mid gSg^{-1}=S\}\).
Page: 123
Chapter: 4
Section: 3
Variables: \(G\) be a group,
\(S\subseteq G\)
Notation: \(G_S\)
Typ: A subgroup of \(G\)

Name: The stabilizer of \(a\)
Definition: \(\{g\in G\mid ga=a\}\).
Page: 051
Chapter: 2
Variables: \(G\) be a group,
\(A\) a set,
\(a\in A\) and
\(\cdot:G\times A\to A\) a group action.
Notation: \(G_a\)
Typ: A subgroup of \(G\)

Name: The stabilizer of \(B\)
Definition: \(\{g\in G\mid gB=B\}\).
Page: 051
Chapter: 2
Variables: \(G\) be a group,
\(A\) a set,
\(B\subseteq A\) and
\(\cdot:G\times A\to A\) a group action.
Notation: \(G_B\)
Typ: A subgroup of \(G\)

Name: A cyclic group
Definition: A group generated by a single element
Page: 54 56
Chapter: 2
Section: 3
Index: 4
Kind: Theorem
Definition2: \(\mathbb Z\) or \(Z_n\) for some \(n\)

Smaller0: \(x^n=x^m=1\)
Greater0: \(x^{\text{gcd}(n,m)}=1\)
Page: 055
Chapter: 2
Section: 3
Variables: \(G\) a group,
\(x\in G\)
\(n,m\in\mathbb N^{>0}\).
Kind: Proposition
Hide middle: yes
Greater1: \(|x|\) divides \(m\)
Connect Greater0: and

Smaller0: \(|x|=\infty\)
Greater0: \(\infty\)
Page: 057
Chapter: 2
Section: 3
Variables: \(G\) be a group,
\(g\in G\)
\(a\in\mathbb Z\setminus\{0\}\).
Index: 5
Kind: Prop
Smaller to greater: iff
Prefix Greater: \(|x^a|=\)

Question: \(|x^a|=\)
Answer: \(\frac{|x|}{\text{gcd}(|x|,a)} \).
Page: 057
Chapter: 2
Section: 3
Variables: \(G\) be a group,
\(g\in G\) and
\(a\in\mathbb Z\setminus\{0\}\).
Index: 5 (3)
Kind: Prop

Question: The subgroups of \(\mathbb Z\) are
Answer: the trivial group and groups isomorphic to \(\mathbb Z\) and in bijection with \(\mathbb N\).
Page: 058
Chapter: 2
Section: 3
Index: 7
Kind: Theorem

Question: Subgroup of \(\mathbb Z/n\mathbb Z\) are
Answer: of the form \(\mathbb Z/a\mathbb Z\) for \(a\) dividing \(n\).
Page: 058
Chapter: 2
Section: 3
Index: 7
Kind: Theorem

Name: The class of subgroups of \(G\)
Page: 062, 85, 94
Section: 4, 1, 2
Chapter: 2, 3, 3
Index: 8, 1, 15
Variables: \(G\) a group.
Kind: prop, ex, Cor
Under: intersection
Condition: arbitrary non-empty
Under2: inverse of a subgroup morphism
Condition2: arbitrary
Under3: product
Condition3: mulitplied by a normal subgroup

Smaller0: \(x\) and \(y\) commutes
Greater0: \(|x||y|\)
Page: 064
Chapter: 2
Section: 4
Variables: \(G\) a group,
\(x,y\in G\)
Prefix Greater: \(|\langle x,y\rangle|=\)
Extra: counter-example ?

Name: A sublattice of \(L\)
Definition: A graph with a subset of nodes of \(L\), and whose edges wich are composition of edges of \(L\), not going through a node of this thing
Page: 070
Chapter: 2
Section: 5
Variables: \(L\) be a lattice.

Name: The fiber of \(\phi\) above \(a\)
Definition: \(\{g\in G\mid \phi(g)=a\}\)
Page: 073, 76
Chapter: 3,
Section: 1,
Variables: \(G,A\) groups
\(\phi:G\to A\) an homomorphism
\(a\in A\).
Notation: \(X_a\)
Definition2: \(\{uk\mid k\in \mathrm{ker}\phi\}\), with \(u\in\phi^{-1}(a)\) fixed
Conjdef: The subset of \(G\), equivalently

Page: 074
Chapter: 3
Section: 1
Name: \(G\) modulo \(K\)
Set: The fibers of \(\phi\) and
Variables: \((G,+_G,0_G),(H,+_H,0_H)\) groups,
\(\phi:G\to H\) an homomorphism
with kernel \(K\).
Notation: \(G/K\)
*: Product of the fiber above \(a\) and above \(b\) is the fiber above \(ab\).
+: Fiber above \(0_H\)
÷: The fiber above \(a\) is sent to the fiber above \(-_Ha\)
Name2: The factor group of \(G\) by \(K\)
Abbreviation: \(G\) mod \(K\)
Name3: The quotient group of \(G\) by \(K\)

Question: For any \(u\in K\) \(uX=\)
Answer: \(X\).
Page: 076
Chapter: 3
Section: 1
Variables: \(G,H\) groups
\(\phi:G\to H\) an homomorphism with kernel \(K\)
\(X\) be a fiber
Index: 2
Kind: Prop

Name: The kernel of \(\phi\)
Definition: \(\phi^{-1}(1_H)\).
Page: 076
Chapter: 3
Section: 1
Variables: \((G,\cdot,1_G),(H,\circ,1_H)\) groups,
\(\phi:G\to H\) an homomorphism.
Index: 2
Kind: Prop
Typ: A subgroup of \(G\)

Name: The image of \(\phi\)
Chapter: 3
Section: 1
Variables: \((G,\cdot,1_G),(H,\circ,1_H)\) groups, \(\phi:G\to H\) an homomorphism.
Typ: A subgroup of \(H\)

Question: Let \(K\) be a kernel, then \(uK\circ vK=\)
Answer: \((uv)K\)
Page: 077
Chapter: 3
Section: 1
Variables: \((G,\cdot,1_G),(H,\circ,1_H)\) groups, \(\phi:G\to H\) an homomorphism with kernel \(K\).
Kind: Theo

Question: The fibers of \(\mathbb Z/n\mathbb Z\) are
Answer: of the form \(a+n\mathbb Z\).
Page: 078
Chapter: 3
Section: 1
Variables: \(n\in\mathbb N^{>0}\)
Kind: Ex

Smaller0: \(\phi\) is an isomorphism
Greater0: is \(\{1_G\}\)
Page: 078
Chapter: 3
Section: 1
Variables: \((G,\cdot,1_G),(H,\circ,1_H)\) groups,
\(\phi:G\to H\) an homomorphism
Index: 2
Kind: Ex
Hide middle: yes
Prefix Greater: \(\phi\)'s kernel
Greater1: singletons
Connect Greater0: and \(\phi\)'s fibers are
Extra: counter-example ? I think there are non concerning fiber

Name: The trivial homomorphism from \(G\) to \(H\)
Definition: [$]\defFun{G}{H}{g}{1}[/$].
Page: 079
Chapter: 3
Section: 1
Variables: \(G,H\) groups
Kind: Ex

Smaller0: \(uN=vN\)
Greater0: \(v^{-1}u\in N\)
Page: 080
Chapter: 3
Section: 1
Variables: \(N\) be any subgroup of a group \(G\). Let \(u,v\in G\).
Index: 4
Kind: Prop
Smaller to greater: iff
Hide middle: yes

Smaller0: \(uN=vN\)
Greater0: \(u\) and \(v\) are representative of the same cosets.
Page: 080
Chapter: 3
Section: 1
Variables: \(N\) be any subgroup of a group \(G\)
\(u,v\in G\).
Index: 4
Kind: Prop
Smaller to greater: iff
Hide middle: yes

Name: The conjugate of \(n\) by \(g\)
Definition: \(gng^{-1}\).
Page: 082
Chapter: 3
Section: 1
Variables: \(G\) a group,
\(n,g\in G\)
Notation: \(n^g\)
Kind: Def

Name: A subgroup of \(G\) normalized by \(g\)
Definition: A subgroup \(N\le G\) such that \(gNg^{-1}=N\).
Page: 082
Chapter: 3
Section: 1
Variables: \(G\) a group
\(g\in G\)
Kind: Def

Name: An element of \(G\) which normalize \(N\)
Definition: An element \(g\in G\) such that \(gNg^{-1}=N\).
Page: 082
Chapter: 3
Section: 1
Variables: \(G\) a group
\(N\le G\)
Kind: Def

Name: The natural projection of \(G\) onto \(G/N\)
Definition: \(g\mapsto gN\).
Page: 083
Chapter: 3
Section: 1
Variables: \(G\) a group
\(N\trianglelefteq G\).
Notation: \(\pi\)
Kind: Def

Name: The complete preimage of \(H\) in \(G\)
Definition: The preimage of \(H\) under the natural projection homomorphism.
Chapter: 3
Section: 1
Variables: \(G\) a group, \(N\trianglelefteq G\) and \(H\le G/N\).
Typ: A subgroup of \(G\) containing \(N\)

Smaller0: \(H\le Z(G)\)
Greater0: \(H\) is normal
Page: 084, 120
Chapter: 3
Section: 1
Variables: \(G\) a group
\(H\le G\).
Index: 2, 5
Kind: Ex, cor
Hide middle: yes
Smaller1: \(H\) is characteristic
Extra: Def 2, 135 et index ?
Connect Smaller0: or
Smaller2: \(H\)'s index is the smallest prime dividing \(|G|\)

Name: A left coset of \(H\) in \(G\)
Definition: \(gH\) for some \(g\in G\).
Page: 77
Chapter: 3
Section: 1
Variables: \(G\) a group
\(H\le G\).
Kind: Def

Name: A right coset of \(H\) in \(G\)
Definition: \(Hg\) for some \(g\in G\).
Page: 77
Chapter: 3
Section: 1
Variables: \(G\) a group
\(H\le G\).
Kind: Def

Name: The index of \(H\)
Definition: The number of left coset of \(H\) in \(G\).
Page: 090
Chapter: 3
Section: 2
Variables: \(G\) a group
\(H\le G\).
Notation: \(|G:H|\)
Kind: Def
Representation: for \([A:B]=n\)

Question: Property of the order of \(x\):
Answer: it divides \(|G|\).
Page: 090
Chapter: 3
Section: 2
Variables: \(G\) a finite group and \(x\in G\).
Index: 9
Kind: Cor

Question: The normal subgroup of \(Q_8\) are
Answer: \(\langle -1\rangle\), \(\langle i\rangle\), \(\langle j\rangle\) and \(\langle k\rangle\).
Chapter: 3
Section: 2

Question: The normal subgroup of \(D_8\) are
Answer: \(\langle r\rangle\), \(\langle r^2\rangle\), \(\langle sr,r^2\rangle\) and \(\langle s,r^2\rangle\).
Chapter: 3
Section: 2

Name: Cauchy's Theorem
Definition: \(G\) has an element of order \(p\).
Page: 093
Chapter: 3
Section: 2
Variables: \(G\) a group an
\(p\) prime dividing \(|G|\).
Index: 11
Kind: Theorem

Name: Sylow's theorem
Definition: \(G\) has a subgroup of order \(p^\alpha\).
Page: 093
Chapter: 3
Section: 2
Variables: \(p\) prime,
\(\alpha,m\in\mathbb N\)
\(G\) a group of order \(p^\alpha m\)
Index: 12
Kind: Theorem
Assuming: \(p\) does not divise \(m\)
Construction: ?

Question: \(H,K\) be subgroups. \(|HK|=\)
Answer: \(\frac{|H||K|}{|H\cap K|} \).
Page: 093
Chapter: 3
Section: 2
Variables: \(G\) a group
\(H,K\le G\).
Index: 13
Kind: Prop

Smaller0: \(H\le N_G(K)\)
Greater0: \(H\cap K\) and \(HK\)
Page: 094
Chapter: 3
Section: 2
Variables: \(G\) a group
\(H,K\le G\).
Index: 15
Kind: Cor
Greater1: a normal subgroup of \(H\)
Connect Greater0: are, respectively
Greater2: a subgroup of \(G\)
Connect Greater1: and

Name: A set which normalizes \(K\)
Definition: \(H\subseteq N_G(K)\)
Page: 094
Chapter: 3
Section: 2
Variables: \(G\) be a group,
\(K\le G\).
Kind: Def
Definition2: \(hKh^{-1}=K\) for each \(h\in H\)
DefType: \(H\subseteq G\)

Name: A set which centralizes \(K\)
Definition: A subset of \(C_G(K)\).
Page: 094
Chapter: 3
Section: 2
Variables: \(G\) be a group and \(K\le G\).
Kind: Def

Smaller0: \(G/\text{ker }\phi\)
Greater0: \(\phi(G)\)
Page: 097
Chapter: 3
Variables: \(G,H\) groups,
\(\phi:G\to H\) an homomorphism.
Index: 16
Kind: Theo
Smaller to greater: is isomorphic to

Smaller0: \(|G:\text{ker }\phi|\)
Greater0: \(|\phi(G)|\)
Page: 097
Chapter: 3
Variables: \(G,H\) groups
\(\phi:G\to H\) an homomorphism.
Index: 17
Kind: Cor
Smaller to greater: equals

Smaller0: injective
Greater0: \(|\text{ker }\phi|=1\)
Page: 097
Chapter: 3
Variables: \(G,H\) groups
\(\phi:G\to H\) an homomorphism.
Index: 17
Kind: Cor
Smaller to greater: iff
Prefix Smaller: \(\phi\) is

Name: Diamond Isomorphism Theorem
Definition: \(AB\le G\)
Page: 097
Chapter: 3
Variables: \(G\) be a group,
\(B\le G\) and
\(A\le N_G(B)\).
Index: 18
Kind: Theo
Definition2: \(B\trianglelefteq AB\)
Conjdef: All of
Definition3: \(A\trianglelefteq AB\)
Name2: Second Isomorphism Theorem
Extra: \(A\trianglelefteq AB\) is trivial and not in the book
Definition4: \(A\cap B\trianglelefteq A\)
Definition5: \(AB/B\) is isomorphic to \(A/(A\cap B)\).

Smaller0: \(A\le N_G(B)\)
Greater0: \(AB\le G\)
Page: 097
Chapter: 3
Variables: \(G\) be a group,
\(A,B\le G\)
Index: 18
Kind: Theo
Greater1: \(B\trianglelefteq AB\)
Extra: \(A\trianglelefteq AB\) is trivial and not in the book
Greater2: \(A\trianglelefteq AB\)
Greater3: \(A\cap B\trianglelefteq A\)

Smaller0: \(A\le N_G(B)\)
Greater0: \(AB/B\)
Page: 097
Chapter: 3
Variables: \(G\) be a group,
let \(A,B\le G\)
Index: 18
Kind: Theo
Smaller1: \(AB\le G\)
Name: Diamond Isomorphism Theorem
Greater1: \(A/(A\cap B)\)
Connect Greater0: is isomorphic to

Q: \([...][...] [...]\).
A: K/H; \trianglelefteq; G/H
Page: 098
Chapter: 3
Variables: \(G\) be a group,
\(H,K\trianglelefteq G\) and
\(H\le K\).
Index: 19
Kind: Theo

Smaller0: \((G/H)/(K/H)\)
Greater0: \(G/K\)
Page: 098
Chapter: 3
Variables: \(G\) be a group
\(H,K\trianglelefteq G\)
\(H\le K\)
Index: 19
Kind: Theorem
Smaller to greater: is isomorphic to
Hide middle: yes
Name: Third Isomorphism Theorem
Name2: Invert and cancel

Smaller0: the set of subgroups \(A\) of \(G\) which contains \(N\)
Greater0: the set of subgroups \(A/N\) of \(G/N\)
Page: 098
Chapter: 3
Variables: \(G\) be a group
\(N\trianglelefteq G\).
Index: 19
Kind: Theo
Smaller to greater: is in bijection with
Name: Fourth Isomorphism Theorem

Smaller0: \(\langle A,B\rangle/N\)
Greater0: \(\langle A/N,B/N\rangle \)
Page: 099
Chapter: 3
Variables: \(G\) be a group
\(N\trianglelefteq G\)
\(N\le A,B\le G\).
Index: 20 (3)
Kind: Theo
Smaller to greater: =
Hide middle: yes
Name: Fourth Isomorphism Theorem
Name2: Lattice Isomorphism Theorem

Smaller0: \((A\cap B)/N\)
Greater0: \(A/N\cap B/N\)
Page: 099
Chapter: 3
Variables: \(G\) be a group
\(N\trianglelefteq G\)
\(N\le A,B\le G\).
Index: 20 (4)
Kind: Theo
Smaller to greater: \(=\)
Name: Fourth Isomorphism Theorem
Name2: lattice Isomorphism Theorem

Smaller0: \(A\trianglelefteq B\)
Greater0: \(A/N\trianglelefteq B/N\)
Page: 099
Chapter: 3
Variables: \(G\) be a group
\(N\le A,B\le G\).
Index: 20 (1)
Kind: Theo
Smaller to greater: iff
Assuming: \(N\trianglelefteq G\)

Smaller0: well defined on \(G/N\)
Greater0: \(N\le\text{ker }\phi\)
Page: 100
Chapter: 3
Variables: \(G,H\) groups
\(N\le G\)
\(\phi:G\to H\).
Smaller to greater: iff
Prefix Smaller: \(\phi\) is

Name: A simple group
Definition: A group whose order is greater than 1 and whose only normal subgroups are 1 and itself.
Page: 102
Chapter: 3
Section: 4
Kind: Def
Example: \(A_5\).

Name: A composition series of \(G\)
Definition: A sequence of subgroups
\[1=N_0\le \dots \le N_k=G\]
such that \(N_i\trianglelefteq N_{i+1}\) and \(N_{i+1}/N_i\) is a simple group
Page: 103
Chapter: 3
Section: 4
Variables: \(G\) a group.
Kind: Def

Name: The Hölder program
Definition: 1- Classify all finite simple groups
2- Find all ways of putting «simple groups together» to form other groups.
Page: 103
Chapter: 3
Section: 4

Smaller0: \(G\) is a simple group
Greater0: \(G\) is isomorphic to \(Z_p\) with \(p>2\) prime
Page: 104
Chapter: 3
Section: 4
Smaller to greater: iff
Smaller1: \(G\)'s order is odd
Extra: Called Feit-Thompson by the book. According to
https://en.wikipedia.org/wiki/Feit–Thompson_theorem this theorem is more general

Name: A solvable group
Definition: has a chain of subgroups \(1=H_0\trianglelefteq \dots \trianglelefteq H_s=G\) such that each \(H_{i+1}/H_i\) is Abelian
Page: 105, 106
Chapter: 3
Section: 4
Index: , 8
Kind: Def, ex
Definition2: such that for each \(n\) dividing [$]\abs{G}[/$], with [$]\left(n,\frac{\abs{G}}{n}\right)=1[/$], \(G\) has a subgroup of order \(n\) (for \(G\) finite)
Conjdef: equivalently
Definition3: has a chain of subgroups \(1=H_0\trianglelefteq \dots \trianglelefteq H_s=G\) such that each \(H_{i+1}/H_i\) is cyclic (for \(G\) finite)
Extra: WHAT ???
Definition4: whose composition factors are of prime order (for \(G\) finite)
Definition5: has a chain of subgroups \(1=H_0\trianglelefteq \dots \trianglelefteq H_s=G\) such that each \(H_{i+1}/H_i\) is Abelian and \(H_i\trianglelefteq G\) (for \(G\) finite)
DefType: A group \(G\)

Smaller0: \(N\) and \(G/N\) are solvable
Greater0: \(G\) is solvable
Page: 105
Chapter: 3
Section: 4
Variables: \(G\) a group
\(N\le G\).
Hide middle: yes
Smaller1: \(N\trianglelefteq G\)
Construction: The construction is exactly the same as the one proving that \(N\) occurs has a composition series of \(G\)

Definition: 1 (respectively -1) if there is an even number of pairs \((i<j)\) such that \(\sigma(j)<\sigma(j)\).
Page: 108
Section: 5
Chapter: 3
Variables: \(n>0\)
\(\sigma\in S_n\).
Notation: \(\epsilon(\sigma)\)
Name: The sign of \(\sigma\)
Morphism: morphism

Name: The sign of an \(n\)-tuple
Definition: \((-1)^{n}\)
Chapter: 3
Section: 5
Variables: \(n>0\).

Name: The alternating group of degree \(n\)
Definition: The set of even permutation.
Page: 107
Chapter: 3
Section: 5
Variables: \(n\in\mathbb N\)
Notation: \(A_n\)

Name: An even permutation
Definition: A permutation whose sign is 1.
Page: 108
Chapter: 3
Section: 5
Variables: \(n\in\mathbb N\)
\(\tau\) a permutation of \(\{1,\dots,n\}\).
Index: 2
Kind: Def

Question: Let \(A=\{1,\dots,n\}\), \(i\in A\) and consider the right action of the group \(S_n\) where \(\sigma\cdot i=\sigma(i)\). Then \(G_i\) is isomorphic to
Answer: \(S_{n-1}\).
Page: 113
Chapter: 4
Section: 1
Kind: Ex

Q: Consider the right action of the group \(D_8\). Then \(\sigma_{ [...] }=[...]\).
A: r; (1 2 3 4)
Page: 113
Chapter: 4
Section: 1
Variables: \(A\) be a square with vertices numeroted 1, 2, 3 and 4
Index: 2
Kind: Ex

Q: Consider the right action of the group \(D_8\). Then \(\sigma_{ [...] }=[...]\).
A: s; (2 4)
Page: 113
Chapter: 4
Section: 1
Variables: \(A\) be a square with vertices numeroted 1, 2, 3 and 4
Index: 2
Kind: Ex

Smaller0: \(G_a\)
Greater0: the subgroup of order 2 generated by the reflection about the line passing through \(a\) and the center of the square
Page: 113
Chapter: 4
Section: 1
Variables: \(A\) be a square with vertices numeroted 1, 2, 3 and 4
\(a\) be a vertice of \(A\).
Consider the right action of the group \(D_8\).
Index: 3
Kind: Ex
Smaller to greater: equals
Hide middle: yes

Name: The permutation afforded by \(\cdot\)
Definition: the associated permutation representation of \(G\).
Page: 114
Chapter: 4
Section: 1
Variables: \(G\) a group,
\(A\) a set.
\(\cdot\) an actiof of \(G\) on \(A\)
Kind: Proposition
Name2: The permutation induced by \(\cdot\)

Smaller0: the actions of \(G\) on \(A\)
Greater0: the homomorphisms of \(G\) into \(S_A\)
Page: 114
Chapter: 4
Section: 1
Variables: \(G\) a group
\(A\) a set
Kind: Proposition
Smaller to greater: are in bijection with
Hide middle: yes

Question: The relation on \(A\) defined by «\(a\sim b\) if and only if \(a=g\cdot b\) for some \(g\)» is
Answer: an equivalence relation.
Page: 114
Chapter: 4
Section: 1
Variables: \(G\) be a group acting on the nonempty set \(A\).
Index: 2
Kind: Proposition

Name: The orbit of \(G\) containing \(a\)
Definition: The equivalence class \(\{g\cdot a\mid g\in G\}\).
Page: 115
Chapter: 4
Section: 1
Variables: \(G\) be a group actiong on the nonempty set \(A\). Let \(a\in A\)
Index: (1)
Kind: Definition

Name: A transitive action of \(G\) on \(A\)
Definition: An action which has a single orbit, i.e., given any two elements \(a,b\in A\) there is some \(g\in G\) such that \(a=g\cdot b\).
Page: 115
Chapter: 4
Section: 1
Variables: \(G\) be a group actiong and \(A\) a set.
Index: (2)
Kind: Definition

Smaller0: \(G\) acts trivially on \(A\)
Greater0: equals \(G\)
Page: 115
Chapter: 4
Section: 1
Variables: \(G\) a group,
\(a\in G\)
Kind: Ex
Hide middle: yes
Prefix Greater: \(G_a\)
Greater1: the orbits are the singletons of elements of \(A\)
Connect Greater0: and

Smaller0: \(G\) acts trivially on \(A\)
Greater0: \(|A|=1\)
Page: 115
Chapter: 4
Section: 1
Kind: Ex
Smaller to greater: iff
Prefix Greater: (
Greater1: this action is transitive
Suffix Greater: )

Name: A group \(G\) with a transitive action on a set \(S\).
A subgroup \(A\le G\), such that the action of \(A\) on \(S\) is not transitive
Page: 115
Chapter: 4
Section: 1
Index: 3
Kind: Ex
Example: Let \(G=S_4\), \(A=\{(12),(34)\}\le G\), acting on \(S=\{1,2,3,4\}\).

Name: A permutation group
Definition: A subgroup of a symmetric group.
Chapter: 4
Section: 1
Kind: Ex

Question: The action of \(G\) on \(A\) is
Answer: transitive.
Page: 119
Chapter: 4
Section: 2
Variables: \(G\) be a group.
\(H\) be a subgroup of \(G\)
\(A\) be the set of left coset of \(H\) in \(G\)
\(\cdot\) the left multiple of \(G\) on \(A\).
Index: 3
Kind: Theorem

Question: \(G\) is isomorphic to
Answer: a subgroup of some symmetric group.
Page: 120
Chapter: 4
Section: 2
Variables: \(G\) a group
Kind: Corollary
Name: Cayley's theorem

Name: Two conjugate elements of \(G\)
Definition: Two elements \(a\) and \(gag^{-1}\) for \(a,g\in G\).
Page: 123
Chapter: 4
Section: 3
Variables: \(G\) a group.

Name: A conjugacy classes of \(G\)
Definition: An orbit of \(G\) acting on itself by conjugation.
Page: 123
Chapter: 4
Section: 3
Variables: \(G\) a group.
Kind: Def

Question: The action by conjugation of \(G\) on itself is
Answer: not transitive.
Page: 123
Chapter: 4
Section: 3
Variables: \(G\) a group.
Index: 1
Kind: Ex
Assuming: \(|G|>1\)

Name: Conjugate subsets
Definition: \(S\) and \(gSg^{-1}\), for some \(g\in G\) and \(S\subseteq G\).
Page: 123
Chapter: 4
Section: 3
Variables: \(G\) a group.

Smaller0: the number of conjugates of \(s\) in \(G\)
Greater0: the index of the centralizer of \(s\)
Page: 123
Chapter: 4
Section: 3
Variables: \(G\) a group
\(s\in G\).
Index: 6
Kind: Proposition
Smaller to greater: is equal to
Hide middle: yes
Construction: \(N_G({s})=C_G(s)\), so this follows from the fact that «the number of conjugate of \(\{s\}\)» equals «the index of the normalizer of \(\{s\}\))

Smaller0: [$]\abs{G}[/$]
Greater0: [$]\abs{Z(G)}+\sum_{i=1}^{r}\abs{G:C_G(g_i)}[/$] where \(g_1,\dots,g_r\) are representative of the distinct conjugacy classes of \(G\) not contained in \(Z(G)\) of \(G\)
Page: 123
Chapter: 4
Section: 3
Variables: \(G\) be a finite group.
Index: 6
Kind: Proposition
Smaller to greater: is equal to
Name: The class equation
Construction: Class equation of the action: conjugation
French: Équation aux classes

Smaller0: [$]\abs{S}[/$]
Greater0: [$]\abs{S_0}+\sum_{i=1}^{r}\frac{\abs{G}}{\abs{G_{s_i}}}[/$] where \(s_1,\dots,s_r\) are elements in each orbit, \(G_{s_i}\) a stabilizer of \(s_i\), and \(S_0\) the set of fixpoint of \(\cdot\)
Page: 123
https://groupprops.subwiki.org/wiki/Class_equation_of_a_group_action
Chapter: 4
Section: 3
Variables: \(G\) be a finite group.
\(S\) a set
\(\cdot:G\times S\to S\) an action of \(G\) on \(S\)
Index: 6
Kind: Proposition
Smaller to greater: is equal to
Name: The class equation of group action
Construction: \(S\) is the union of the orbits \((\mathcal O_i)_i\). [$]\defFun{G/G_x}{\mathcal O_x}{g}{g\cdot x}[/$] is a bijection

Smaller0: \(\langle g\rangle\)
Greater0: \(C_G(g)\)
Chapter: 4
Section: 3
Variables: \(G\) a group and \(g\in G\).
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \(G\)'s order is the power of a prime
Greater0: \(G\) has a nontrivial center
Page: 125
Chapter: 4
Section: 3
Variables: \(G\) a group.
Index: 8
Kind: Theorem
Hide middle: yes

Smaller0: \(|G|=p^2\) with \(p\) prime
Greater0: \(G\) is either \(Z_{p^2}\) or \(Z_p\times Z_p\)
Page: 125
Chapter: 4
Section: 3
Variables: \(G\) a group.
Index: 9
Kind: Corollary
Smaller to greater: iff
Hide middle: yes
Context: Group

Smaller0: \(\tau\sigma\tau^{-1}\)
Greater0: \((\tau(a_{0,0})\dots\tau(a_{0,i_0}))\dots(\tau(a_{j,0})\dots \tau(a_{j,i_j}))\)
Page: 125
Chapter: 4
Section: 3
Variables: \(S\) a set,
\(\tau,\sigma\) permutations with
\(\sigma=(a_{0,0}\dots a_{0,i_0})\dots(a_{j,0}\dots a_{j,i_j})\)
Index: 10
Kind: Proposition
Smaller to greater: equals

Name: The cycle type of \(\sigma\)
Definition: The integers \(i_1,\dots,i_r\), assimng \(i_j\le i_{j+1}\).
Page: 126
Chapter: 4
Section: 3
Variables: \(\sigma=(a_{0,0}\dots a_{0,i_0})\dots(a_{j,0}\dots a_{j,i_j})\).
Index: 1
Kind: Definition

Q: [...] is [...].
A: A partition of \(n\); any nondecreasing sequence of positive integers whose sum is \(n\)
Page: 126
Chapter: 4
Section: 3
Variables: \(n\in\mathbb Z^+\)
Index: 2
Kind: Definition

Smaller0: the cycle type of \(\sigma\) is \(1,\dots,1,m\) where the \(m\) is preceded by \(n-m\) ones
Greater0: \(\sigma\) is an \(m\)-cycle
Page: 126
Chapter: 4
Section: 3
Variables: \(1\le n\le m\)
\(\sigma\in S_n\).
Smaller to greater: iff

Smaller0: \(\tau\) and \(\sigma\) are conjugate in \(S_n\)
Greater0: \(\tau\) and \(\sigma\) have the same cycle type
Page: 126
Chapter: 4
Section: 3
Variables: \(n\in\mathbb N\)
\(\tau,\sigma\in S_n\).
Index: 11
Kind: Proposition
Smaller to greater: iff

Question: In \(D_{2n}\), the order of \(s\) is
Answer: 2.
Page: 025
Chapter: 1
Section: 2
Variables: \(n\in\mathbb N\)
Index: (2)

Question: The stabilizer in \(G\) of the point \(1H\) is
Answer: the subgroup \(H\)
Page: 119
Chapter: 4
Section: 2
Variables: \(G\) be a group
\(H\le G\)
\(G\) act by left multiplication on \(G/H\)
Index: 3
Kind: Theo

Question: The kernel of the the left action \(g\cdot (g'H)= (gg')H\) is equal to
Answer: \(\bigcap _{x\in G} xHx^{-1}\)
Page: 119
Chapter: 4
Section: 2
Variables: \(G\) be a group
\(H\le G\)
Index: 3
Kind: Theo

Smaller0: \(\text{ker}\pi_H\)
Greater0: the largest normal subgroup of \(G\) contained in \(H\)
Page: 119
Chapter: 4
Section: 2
Variables: \(G\) be a group,
\(H\) be a subgroup of \(G\) and
\(G\) act by \(\pi_H\), the left multiplication on the set \(A\) of left cosets of \(H\) in \(G\).
Index: 3 (3)
Kind: Theo
Smaller to greater: equals
Smallest: \(\bigcap _{x\in G} xHx^{-1}\)

Smaller0: \(H\) is a normal subgroup
Greater0: \(K\) is either included in \(H\) or disjoint from it
Page: 127
Chapter: 4
Section: 3
Variables: \(G\) a group,
\(H\subseteq G\).
\(K\) is a conjugacy class of \(G\)
Index: -3
Kind: line
Hide middle: yes

Definition: \(b^{-1}ab\)
Page: 129
Chapter: 4
Section: 3
Variables: \(G\) a group
\(a,b\in G\).
Notation: \(a^b\)

Question: Example of right action on \(G\)
Answer: \(b\mapsto a\cdot{}b=b^{-1}ab\)
Page: 129
Chapter: 4
Section: 3
Variables: \(G\) a group.

Name: Corresponding group actions
Definition: \(\circ:A\times G\to A\) and \(\cdot:G\times A\to A\) such that for all \(a\in A\), \(g\in G\), \(a\cdot g=g^{-1}\circ a\).
Page: 129
Chapter: 4
Section: 3
Variables: \(G\) a group
\(A\) a set

Name: An automorphism of \(G\)
Definition: An isomorphism \(G\to G\)
Page: 133
Chapter: 4
Variables: \(G\) a group
Example: [$]\defFun{\mathbb G}{\mathbb G}{g}{g^k}[/$] with \(k\in\mathbb Z\) for \(G\) Abelian

Definition: whose element are automorphism of \(G\)
Page: 133
Chapter: 4
Variables: \(G\) a group.
Notation: \(\text{Aut}(G)\)
Definition2: whose operation is the composition of homomorphism
Conjdef: The group

Name: Action of \(G\) on \(H\) by conjugation
Definition: [$]\defFun{H}{H}{h}{ghg^{-1}}[/$].
Page: 133
Chapter: 4
Variables: \(G\) a group,
\(g\in G\) and
\(H\le G\).
Index: 13
Kind: Prop
Typ: An automorphism of \(H\)
Assuming: \(H\trianglelefteq G\)

Question: \(gKg^{-1}\) is isomorphic to
Answer: \(K\).
Page: 134
Chapter: 4
Variables: \(G\) a group, \(g\in G\) and \(K\le G\).
Index: 14
Kind: Cor

Question: \(N_G(H)/C_G(H)\) is isomorphic to
Answer: a subgroup of \(\text{Aut}(H)\).
Page: 134
Chapter: 4
Variables: \(G\) a group
\(H\le G\)
Index: 15
Kind: Cor
Construction: \(H\trianglelefteq N_G(H)\) thus \(N_G(H)\) acts by conjugation on \(H\) with \(C_G(H)\) the kernel

Question: \(G/Z(G)\) is isomorphic to
Answer: \(\text{Inn}(G)\).
Page: 134
Chapter: 4
Variables: \(G\) a group
Index: 15
Kind: Cor

Name: An inner automorphism
Definition: A function of the form \(x\mapsto gxg^{-1}\) for some \(g\in G\).
Page: 134
Chapter: 4
Variables: \(G\) a group.
Kind: Def

Name: The set of inner automorphism of \(G\).
Page: 134
Chapter: 4
Variables: \(G\) a group
Kind: Def
Abbreviation: \(\text{Inn}(G)\)

Name: A group \(G\) and an Abelian subgroup which is not included in \(Z(G)\)
Page: 134
Chapter: 4
Index: 1
Kind: Ex
Example: \(A_4\) and its subgroup of pair of permutations.

Name: The Klein 4-group
Definition: \(1\) the identity,
Page: 068
Chapter: 2
Section: 5
Definition2: \(xx=1\),
Conjdef: \(\{1,a,b,c\}\), with
Definition3: \(xy=z\)
Name2: Viergruppe

Name: An abelian subgroup of order 4, distinct from \(\mathbb Z_4\)
Page: 068
Chapter: 2
Section: 5
Example: The Klein 4-group.

Question: \(\text{Inn}(Q_8)\simeq \)
Answer: \(V_4\).
Page: 134
Chapter: 4
Index: 2
Kind: Ex

Question: \(\text{Inn}(D_8)\simeq \)
Answer: \(V_4\).
Page: 134
Chapter: 4
Index: 3
Kind: Ex

Question: \(Z(D_8)= \)
Answer: \(\langle r^2\rangle\).
Page: 134
Chapter: 4
Index: 3
Kind: Ex

Question: \(Z(Q_8)= \)
Answer: \(\langle -1\rangle\).
Page: 134
Chapter: 4
Index: 2
Kind: Ex

Question: \(Z(S_n)= \)
Answer: \(1\).
Page: 134
Chapter: 4
Variables: \(n\ge3\).
Index: (4)
Kind: Ex

Question: \(\text{Inn}(S_n)= \)
Answer: \(S_n\).
Page: 134
Chapter: 4
Variables: \(n\ge3\).
Kind: Ex

Name: \(H\) is a characteristic subgroup of \(G\)
Page: 135
Chapter: 4
Variables: \(G\) a group.
\(H\le G\)
Notation: \(H\) char \(G\)
Kind: Def

Name: A characteristic subgroup of \(G\)
Definition: A subgroup \(H\) such that for every \(\sigma\in\text{Aut}(G)\), \(\sigma(H)=H\).
Page: 135, https://fr.m.wikipedia.org/wiki/Sous-groupe_caractéristique
Chapter: 4
Variables: \(G\) a group.
Kind: Def
Example: 1, G and every subgroup which are the only subgroup of their order
Example2: center of the group
Typ: transitive, normal
Example3: the derived subgroup
Example4: all subgroup
Applied to4: cyclic group

Smaller0: \(K\) char \(H\)
Greater0: \(K\trianglelefteq G\)
Page: 015
Chapter: 4
Variables: \(K\le H\le G\) groups.
Index: (3)
Hide middle: yes
Smaller1: \(H\trianglelefteq G\)
Connect Smaller0: and

Question: The automorphism group of \(\mathbb Z/p^n\mathbb Z\) is
Answer: \(\mathbb Z/\left(p^{n-1}(p-1)\right)\mathbb Z\).
Page: 136
Chapter: 4
Variables: \(p>2\) odd
\(n\in\mathbb Z^+\)
Index: 17 (1)
Kind: Prop

Question: The automorphism group of \(\mathbb Z/2^n\mathbb Z\)
Answer: \(Z_2\times Z_{2^{n-2} }\).
Page: 136
Chapter: 4
Variables: \(n\ge 3\).
Index: 17 (2)
Kind: Prop
Example: Not cyclic

Question: The automorphism of the vector space \((\mathbb F_p)^n\)
Answer: The nonsingular linear transformation from \(V\) to itself.
Page: 136
Chapter: 4
Variables: \(p\) prime and \(n\in\mathbb Z^+\).
Index: 17 (3)
Kind: Prop
Extra: \(GL_n(\mathbb F_p)\)

Question: The automorphism group of \(Q_8\) is
Answer: \(S_4\).
Page: 136
Chapter: 4
Index: 17 (5)
Kind: Prop

Name: A \(p\)-group
Definition: A group whose's order is a power of \(p\).
Page: 139
Chapter: 4
Section: 5
Variables: \(p\) prime.
Index: (1)
Kind: Def

Name: A \(p\)-subgroup
Definition: A subgroup \(H\le G\) whose order is a power of \(p\).
Page: 139
Chapter: 4
Section: 5
Variables: \(G\) a group
\(p\) prime
Index: (1)
Kind: Def

Name: A Sylow \(p\)-subgroup of \(G\)
Definition: A subgroup of \(G\) whose index is coprime with \(p\).
Page: 139
Chapter: 4
Section: 5
Variables: \(G\) a group
\(p\) prime
Index: (2)
Kind: Def

Name: The set of Sylow \(p\)-subgroup of \(G\)
Page: 139
Chapter: 4
Section: 5
Variables: \(G\) a group
\(p\) prime.
Notation: \(Syl_p(G)\)
Index: (3)
Kind: Def

Question: \(n_p(G)\)
Answer: \(\equiv 1 \mod p\),
Page: 139
Chapter: 4
Section: 5
Variables: \(G\) a group
\(p\) prime dividing the order of \(G\)
Answer2: divides \(|G|\)
Index: 18 (3)
Kind: Theoren
Name: Sylow's Theorem

Smaller0: \(H\in\text{Syl}_p(G)\)
Greater0: \(K\le gHg^{-1}\) for some \(g\in G\)
Page: 139, 140
Chapter: 4
Section: 5
Variables: \(G\) a group
\(p\) prime dividing the order of \(G\)
\(H,K\le G\).
Index: 18 (2), 19
Kind: Theo, Lem
Smaller1: \(K\) is a \(p\)-subgroup
Name: Sylow's Theorem
Greater1: \(K\cap H\)
Connect Greater0: and \(K\cap N_G(H)=\)

Smaller0: \(P\) is the unique Sylow \(p\)-subgroup of \(G\)
Greater0: \(P\) is characteristic
Page: 142
Chapter: 4
Section: 5
Variables: \(G\) a group
\(p\) prime dividing the order of \(G\)
\(P\le G\)
Index: 20
Kind: Cor
Smaller to greater: iff
Assuming: \(P\) is a Sylow \(p\)-subgroup of \(G\)
Smallest: \(P\) is normal
Greatest: for each set \(X\subseteq G\) whose elements' order is a power of \(p\), it follows that \(\langle X\rangle\) is a \(p\)-group

Smaller0: \(p\) does not divide the order of \(G\)
Greater0: \(\{1\}\)
Page: 142
Chapter: 4
Section: 5
Variables: \(G\) a group and \(p\) prime.
Index: (1)
Kind: Ex
Smaller to greater: iff
Prefix Greater: \(Syl_p(G)\) equals

Smaller0: \(|G|=p^\alpha\)
Greater0: \(\{G\}\)
Page: 142
Chapter: 4
Section: 5
Variables: \(G\) a group
\(p\) prime.
Index: (1)
Kind: Ex
Smaller to greater: iff
Prefix Greater: \(Syl_p(G)\) is

Name: The \(p\)-primary component of \(G\)
Definition: The set of \(x\in G\) whose order is a power of \(p\)
Page: 142
Chapter: 4
Section: 5
Variables: \(G\) a group
\(p\) prime.
Index: (2)
Kind: Ex
Assuming: \(G\) is Abelian

Smaller0: \(G\) is Abelian
Greater0: the singleton \(\{p\text{-primary component}\}\)
Page: 142
Chapter: 4
Section: 5
Variables: \(G\) a group and \(p\) prime.
Index: (2)
Kind: Ex
Smaller to greater: iff
Hide middle: yes
Prefix Greater: \(Syl_p(G)\) is

Question: The Sylow \(p\)-subgroups of \(S_3\) are:
Answer: -\(\langle(12)\rangle\), \(\langle(23)\rangle\) and \(\langle(13)\rangle\) for \(p=2\) and
-\(\langle(123)\rangle=A_3\) for \(p=3\)
Page: 142
Chapter: 4
Section: 5
Index: (3)
Kind: Ex

Question: The Sylow \(p\)-subgroups of \(A_4\) are:
Answer: -\(\langle(12)(34),(13)(24)\rangle\), for \(p=2\) and
-\(\langle(123)\rangle\), \(\langle(124)\rangle\),\(\langle(134)\rangle\) and \(\langle(234)\rangle\) for \(p=3\)
Page: 142
Chapter: 4
Section: 5
Index: (4)
Kind: Ex

Smaller0: the number of elements in the equivalence class containing \(a\) (i.e. \(\mathcal C_a=\{g\cdot a\mid g\in G\}\))
Greater0: \(|G:G_a|\) (i.e. the index of the stabilizer of \(a\))
Page: 114
Chapter: 4
Section: 1
Variables: \(A\) nonempy set
\(G\) be a group acting on\(A\)
\(a\in A\).
Index: 2
Kind: Proposition
Smaller to greater: is equal to
Construction: [$]\defFun{G/G_a}{\mathcal C_a}{gG_a}{g\cdot a}[/$] is a bijection,

Definition: The number of Sylow \(p\)-subgroup of \(G\).
Page: 139
Chapter: 4
Section: 5
Variables: \(G\) a group and \(p\) prime.
Notation: \(n_p(G)\) or \(n_p\)
Index: (3)
Kind: Def

Question: The number of identity is
Answer: at most one.
Page: 017
Chapter: 1
Variables: \(C\) a set, \(\star:C\times C\to C\).

Smaller0: \(n\ne 4\)
Greater0: simple
Page: 149
Chapter: 4
Section: 6
Variables: \(n\in\mathbb N\)
Index: 24
Kind: Theo
Smaller to greater: iff
Hide middle: yes
Prefix Greater: \(A_n\) is

Question: \(|\Pi_{i\in I}G_i|\) is
Answer: \(\Pi_{i\in I}|G_i|\).
Page: 152
Chapter: 5
Section: 1
Variables: \(I\) a set
\((G_i,\star_i)_{i\in I}\) a class of groups indexed by \(I\)
Kind: Prop

Definition: The subgroup of \(G\) of elements sending everything apart from \(i\) to the identy.
Page: 154
Chapter: 5
Section: 1
Variables: \(I\) a set, \((G_i,\star_i)_{i\in I}\) a class of groups indexed by \(I\) and \(i\in I\), \(G=\Pi_{i\in I}G_i\).
Notation: \(G_i\subseteq G\)
Index: 2 (1)
Kind: Prop

Page: 154
Chapter: 5
Section: 1
Variables: \(I\) a set,
\((G_i,\star_i)_{i\in I}\) a class of groups indexed by \(I\) and
\(i\in I\),
\(G=\Pi_{i\in I}G_i\).
Notation: \(G_i\subseteq G\)
Index: 2 (1)
Kind: Prop
Typ: A subgroup

Question: \(G/G_j\) is isomorphic to
Answer: \(\Pi_{j\in I\setminus\{i\} }G_j\).
Page: 154
Chapter: 5
Section: 1
Variables: \(I\) a set
\((G_i,\star_i)_{i\in I}\) a class of groups indexed by \(I\)
\(j\in I\)
\(G=\Pi_{i\in I}G_i\).
Index: 2 (1)
Kind: Prop

Definition: The function sending a \(I\)-tuple to its \(i\)-th component .
Page: 154
Chapter: 5
Section: 1
Variables: \(I\) a set,
\((G_i,\star_i)_{i\in I}\) a class of groups indexed by \(I\)
\(j\in I\)
\(G=\Pi_{i\in I}G_i\).
Notation: \(\pi_i\)
Index: 2 (2)
Kind: Prop

Definition: [$]\defFun{\Pi_{i\in I}G_i}{G_i}{(g_i)_{i\in\mathbb N}}{g_i}[/$]
Page: 154
Section: 1
Chapter: 5
Variables: \(I\) a set
\((G_i,\star_i)_{i\in I}\) a class of groups indexed by \(I\)
\(j\in I\)
Notation: \(\pi_i\)
Index: Prop
Kind: 2 (2)
Jective: A surjective homomorphism

Question: \(\ker\pi_i\) is
Answer: isomorphic to \(\Pi_{j\in I\setminus\{i\} }G_i\).
Page: 154
Chapter: 5
Section: 1
Variables: \(I\) a set
\((G_i,\star_i)_{i\in I}\) a class of groups indexed by \(I\) and \(j\in I\)
\(G=\Pi_{i\in I}G_i\)
Index: 2 (2)
Kind: Prop

Question: \((x_1\dots x_n)^k=\)
Answer: \(x_1^k\dots x_n^k\).
Page: 155
Chapter: 5
Section: 1
Variables: \(I\) a set
\((G_i,\star_i)_{i\in I}\) a class of groups indexed by \(I\)
\(G=\Pi_{i\in I}G_i\)
\(x_i\in G_i\subseteq G\) for \(1\le i\le n\)
\(k\in\mathbb N\).
Index: (1)
Kind: Ex

Question: \(|x_1\dots x_n|=\)
Answer: \(\text{lcm}(|x_1|,\dots, |x_n|)\).
Page: 155
Chapter: 5
Section: 1
Variables: \(I\) a set,
\((G_i,\star_i)_{i\in I}\) a class of groups indexed by \(I\),
\(G=\Pi_{i\in I}G_i\)
\(x_i\in G_i\subseteq G\) for \(1\le i\le n\).
Index: (1)
Kind: Ex

Name: Elementary Abelian group of order \(p^n\)
Definition: \((Z_p)^n\).
Page: 155
Chapter: 5
Section: 1
Variables: \(p\) prime
\(n\in\mathbb N\)
Notation: \(E_{p^n}\)
Index: (2)
Kind: Ex

Question: The number of subgroups of order \(p\) of \(E_{p^2}\) is
Answer: \(p+1\)
Page: 155
Chapter: 5
Section: 1
Variables: \(p\) prime
Index: (3)
Kind: Ex

Name: A finitely generated group
Definition: A group \(G\) such that there exists a finite subset \(A\) such that \(\langle A\rangle=G\).
Page: 158
Chapter: 5
Section: 2
Index: 1
Kind: Def

Name: The free Abelian group of rank \(r\)
Page: 158
Chapter: 5
Section: 2
Variables: \(r\in\mathbb N\)
Notation: \(\mathbb Z^r\)
Kind: Def

Name: Invariant factor decomposition of an Abelian group
Definition: An Abelian group presented as \(\mathbb Z^r\times\Pi_{i=0}^{s}\mathbb Z_{n_i}\).
Page: 158
Chapter: 5
Section: 2
Index: 3 (1)
Kind: Theo
Assuming: \(2\le n_i\), \(n_{i+1}|n_i\).

Name: Free rank of \(G\)
Definition: \(r\).
Page: 158
Chapter: 5
Section: 2
Variables: \(r,s\in\mathbb N\),
\(2\le n_i\),
\(n_{i+1}|n_i\).
\(G=\mathbb Z^r\times Z_{n_1}\dots\times Z_{n_s}\).
Index: 3 (1)
Kind: Theo
Name2: Betti number of \(G\)

Name: The invariant factors of \(G\)
Definition: \(n_1,\dots,n_s\).
Page: 159
Chapter: 5
Section: 2
Variables: \(r,s\in\mathbb N\),
\(2\le n_i\),
\(n_{i+1}|n_i\).
\(G=\mathbb Z^r\times Z_{n_1}\dots\times Z_{n_s}\).
Kind: Def

Smaller0: 0
Greater0: \(G\) is finite
Page: 158
Chapter: 5
Section: 2
Variables: \(G\) a finitely generated group
Kind: Def
Smaller to greater: iff
Prefix Smaller: the free rank of \(G\) is

Name: An elementary divisor of \(G\)
Definition: Some \(p_i^{\beta_{i,j}}\).
Page: 161
Chapter: 5
Section: 2
Variables: \((p_i)_{1\le i\le k}\) prime numbers,
\((\beta_{i,j})_{1\le j\le t_i}\) non-decreasing sequence of non-negative integers
\(G=\Pi_{1\le i\le k}\Pi_{1\le j\le t_i}Z_{p_i^{\beta_{i,j}}}\)
Kind: Def

Smaller0: \((m,n)=1\)
Greater0: \(Z_{mn} \)
Page: 163
Chapter: 5
Section: 2
Variables: \(m,n\in\mathbb N\)
Index: 6 (1)
Kind: Prop
Smaller to greater: iff
Prefix Greater: \(Z_m\times Z_n\equiv \)

Question: \(Z_n\equiv\)
Answer: \(\Pi_{Z_{p_i^{\alpha_i}}}\).
Page: 163
Chapter: 5
Section: 2
Variables: \(n=\Pi_{i=1}^k{p_i}^{\alpha_i}\)
Index: 6 (1)
Kind: Prop

Name: The Heisenberg group over \(R\)
Definition: The subgroup with elements of the form \(\left(\begin{array}{lll}1&a&b\\ 0&1&c\\0&0&1\end{array}\right)\)
Page: 35
Chapter: 1
Section: 4
Variables: \((R,+,\times)\) a ring,

Name: A maximal subgroup
Definition: A subgroup \(H< G\) such that there exists no \(K\) such that \(H<K<G\).
Page: 65
Chapter: 2
Section: 4
Variables: \((G,\cdot, 1)\) a group
Index: 16
Kind: Ex

Name: A divisible nontrivial abelian group
Definition: A group \((G,.,1)\) such that, for each \(a\in A\), for each \(k\in\mathbb Z\) there is an element \(x\in A\) such that \(x^k=a\).
Page: 66
Chapter: 2
Section: 4
Index: 19
Kind: Ex

Page: 71
Chapter: 2
Section: 5
Name: The quasidihedral group of order \(2^n\)
Variables: \(n\in\mathbb N\)
Kind: Ex
Index: 11
Base: \(\{\sigma,\tau\}\)
Name2: The semihedral group of order \(2^n\)
Relation: \(\sigma ^{2^{n-1} }\)
Extra: The book only consider \(n=4\), generalization thanks to wp
Assuming: \(n\ge 4\)
Relation2: \(\tau^2\)
Relation3: \(\tau\sigma\tau=\sigma^{2^{n-2}-1}\)

Name: The modular group of order 16
Definition: \(\langle u,v\mid u^2=v^8=1, vu=uv^5\rangle\).
Page: 72
Chapter: 2
Section: 5
Index: 14
Kind: Ex
Extra: As far as I understand using wp, there is no simpler definition, no standard name and the name is not generalized for other order

Name: A Hall subgroup of \(G\)
Definition: its index in \(G\) is relatively prime to its order.
Page: 101
Chapter: 3
Variables: \(\langle G,.,1\rangle\) a group
Index: 10
Kind: Ex
Definition2: \((|G:H|,|H|)=1\).
Conjdef: A subgroup \(H\le G\) such that

Name: A block for \(\sigma\)
Definition: A nonempty subset \(B\subseteq A\) such that for all \(\sigma\in G\), either \(\sigma(B)=B\) or \(\sigma(B)\cap B=\emptyset\).
Page: 117
Chapter: 4
Section: 3
Variables: \(A\) a finite set,
\(G\) a transitive permutation group on \(A\),
Index: 7
Kind: Ex

Name: A primitive transitive permutation group on \(A\)
Definition: A permutation group on \(A\) such that every blocks are trivial.
Page: 117
Chapter: 4
Section: 3
Variables: \(A\) a finite set,
Index: 7
Kind: Ex

Name: A doubly transitive permutation group on \(A\)
Definition: A transitive permutation group on \(A\) such that for each \(a\in A\), \(G_a\) is transitive on the set \(A-\{a\}\).
Page: 117
Chapter: 4
Section: 3
Variables: \(A\) a set
Index: 8
Kind: Ex

Name: The \(HK\) double coset of \(x\ G\)
Definition: \(HxK\).
Page: 117
Chapter: 4
Section: 3
Variables: \(G \) a group,
\(H,K\le G\),
\(x\in G\)
Index: 10
Kind: Ex

Name: A fixed point free element
Definition: An element \(\sigma\in G\) such that \(\sigma (a)\ne a\) for each \(a\in A\).
Page: 132
Chapter: 4
Section: 3
Variables: \(A\) a set,
\(G\) a transition group,
Index: 8
Kind: Ex

Name: The direct product of the \(G_i\)'s
Definition: \(\Pi_{i\in I}G_i\).
Page: 152 ,157
Chapter: 5
Section: 1
Variables: \(I\) a non empty index set,
\((G_i,\star _i,1_i)_{i\in I}\) groups
Index: 15
Kind: Def, Ex
Definition2: \(((a_i)_{i\in I},(\Pi_i b_i)_{i\in I})\mapsto (a_i\star_i b_i)_{i\in I}\)
Conjdef: The tuple
Definition3: \((1_i)_{i\in I}\)
Typ: Group

Name: Their restricted direct product
Definition: The restriction of the direct product where there is only a finite number of non-unit elements.
Page: 158
Chapter: 5
Section: 1
Variables: \(I\) a non empty index set,
\((G_i,\star _i,1_i)_{i\in I}\) groups
Index: 17
Kind: Ex
Name2: Their direct sum
Typ: Group

Question: Exemple of group of order \(m\) with no subgroup of order \(n\)
Answer: The group of permutation of the regular tetrahedron is of order 12, and it has no subgroup of cardinal 6
Page: 89
Chapter: 3
Section: 2
Variables: \(n|m\) integers
Index: 8
Kind: Theo

Question: The order of \(A_n\)
Answer: \(n!/2\)
Page: 107
Chapter: 3
Section: 5
Variables: \(n\) integers

Name: An odd permutation
Definition: A permutation whose sign is -1
Page: 108
Chapter: 3
Section: 5
Variables: \(n\in\mathbb N\)
\(\tau\) a permutation of \(\{1,\dots,n\}\).
Index: 2
Kind: Def

Name: A transposition
Definition: A 2-cycle
Page: 107
Chapter: 3
Section: 5

Name: The action by conjugation of \(G\) on itself
Definition: \(g\mapsto (n\mapsto gng^{-1})\).
Page: 052
Chapter: 2
Variables: \(G\) a group
Kind: Def

Name: The cyclic group of order \(q\)
Definition: \(\mathbb Z/q\mathbb Z\)
Page: 56, 1
Chapter: 2,1(Generators and relations for discrete groups)
Section: 3, 1
Variables: \(q\in\mathbb N\)
Notation: \(Z_q\)
Kind: Notation

Definition: The finite field of order \(n\), \(\mathbb Z/n\mathbb Z\)
Page: 34,Frequently Used Notation
Chapter: 1
Section: 4
Variables: \(n\in\mathbb N\)
Notation: \(\mathbb F_n\)

Question: The number of inverse of \(a\) is
Answer: 1
Page: 17
Variables: \((G,\star)\) a group and
\(a\in G\).
Context: Group
Kind: Def

Name: An inverse of \(a\)
Definition: An element \(i\) such that \(a i=ia=1\), where \(1\) is the identity.
Page: 017
Chapter: 1
Variables: \((G,\star)\) a group and
\(a\in G\).
Notation: \(a^{-1}\)
Kind: Def

Question: Example of set generating \(\mathbb Z\)
Answer: \(\{1\}\)
Page: 26
Chapter: 1
Section: 2

Name: The centralizer of \(A\) in \(G\)
Definition: The set of elements commuting with all elements of \(A\).
Page: 049, wp
Chapter: 2
Variables: \(G\) a group
\(A\) a nonempty subset of \(G\).
Notation: \(C_G(A)\)
Kind: Def
Definition2: \(\{g\in G\mid ga=ag\text{ for all }a\in A\}\)
Name2: The commutant of \(A\) in \(G\)
Extra: Subgroup is straightforward
Typ: A normal subgroup of \(G\)

Smaller0: \(|\langle x\rangle|\)
Greater0: \(|x|\)
Page: 55
Chapter: 2
Section: 3
Variables: \(G\) a group,
\(x\in G\)
Kind: Prop
Smaller to greater: =

Smaller0: \(x^i\)
Greater0: \(x^j\)
Page: 55
Chapter: 2
Section: 3
Variables: \(G\) a group,
\(x\in G\) and
\(0\le i<j<|x|\)
Kind: Prop
Smaller to greater: \(\ne\)
Hide sides: 1

Question: The number of infinite families of simple group ?
Answer: 18
Page: 104
Chapter: 3
Section: 4

Question: The number of sporadic simple group ?
Answer: 26
Page: 104
Chapter: 3
Section: 4

Name: A sporadic simple group
Definition: A group which does not belong to the infinite families of simple group.
Page: 104
Chapter: 3
Section: 4

Question: The right action of the group \(S_n\) on \(\{1,\dots,n\}\) where \(\sigma\cdot i=\sigma(i)\) is
Answer: faithful and transitive
Page: 113
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\)
Kind: Ex
Extra: Transitive not in the book

Smaller0: \(n\) is square-free
Greater0: 1
Page: 160
Chapter: 5
Section: 2
Variables: \(n\in\mathbb N\)
Index: 4
Kind: Cor
Prefix Greater: the number of groups of order \(n\) is

Smaller0: \(n\) is prime
Greater0: \(G\) has a subgroup isomorphic to \(Z_{nm}\) with \(m\in\mathbb N\)
Page: 158
Chapter: 5
Section: 2
Variables: \(G\) a finite Abelian group,
\(n\in\mathbb N\)
Smaller1: \(n\) divides \(|G|\)

Question: The number of elementary divisor decomposition of \(G\) is
Answer: exactly one.
Page: 161
Chapter: 5
Section: 2
Variables: \(G\) a finite Abelian group
Kind: Theo

Name: An elementary divisor decomposition of \(G\)
Definition: A sequence \((p_i)_{1\le i\le k}\) of prime numbers,
Sequences \((\beta_{i,j})_{1\le j\le t_i}\) non-decreasing sequence of non-negative integers
Such that \(G=\Pi_{1\le i\le k}\Pi_{1\le j\le t_i}Z_{p_i^{\beta_{i,j} } }\).
Page: 161
Chapter: 5
Section: 2
Variables: \(n=\Pi_{i=1}^k p_i^{\alpha_i}\) with \(p_i\) prime,
\(\alpha_i\in\mathbb N\) and
\(G\) a group of order \(n\)
Kind: Theo

Question: \(t\) can be canonically written as
Answer: a sequence of permutations \((a_1\dots a_{m_1})\dots(a_{m_{k-1}+1},\dots,a_{m_k})\), with \(a_i\ne a_j\) and all \(a_{m_i}\) smaller than all successive elements.
Page: 029
Chapter: 1
Section: 3
Variables: \(n\in\mathbb N\)
\(t\) a permutation of \(\{1,\dots,n\}\).

Definition: The set of inversible elements of \(\mathbb Z/n\mathbb Z\) with the multiplicative function
Page: 017
Chapter: 1
Notation: \((\mathbb Z/n\mathbb Z)^\times\)
Index: 5
Kind: Ex

Name: The lattice of its subgroups
Definition: Nodes are the subgroups and
Page: 066
Chapter: 2
Section: 5
Variables: \(G\) be a group.
Definition2: Whose vertex go from \(A\) to \(B\) if \(A\le B\) and there are no \(C\) such that \(A\le C\le B\) with label the index of the subgroup.
Conjdef: The graph whose
Denoted by: \(L\)

Name: A composition factor
Definition: \(N_{i+1}/N_i\)
Page: 103
Chapter: 3
Section: 4
Variables: \(G\) a group and
\(1=N_0\le \dots \le N_k=G\) a composition series.
Kind: Def

Name: The join of \(A\) and \(B\)
Definition: The smallest subgroup containing \(A\) and \(B\)
Page: 066
Chapter: 2
Section: 5
Variables: \(G\) a group
\(L\) the lattice of subgroup of \(L\)
\(A,B\in L\)
Definition2: The lowest graph above \(A\) and \(B\).

Smaller0: \(G\) is finitely generated
Greater0: \(\mathbb Z^r\times Z_{n_1}\dots\times Z_{n_s}\) such that \(r\in\mathbb N\), the \(n_j\ge2\) and \(n_{i+1}|n_i\) for \(1\le i<s\)
Page: 158
Chapter: 5
Section: 2
Variables: \(G\) a group
Index: 3 (1)
Kind: Theo
Smaller to greater: iff
Prefix Greater: \(G\) is of the form
Name: Fundamental Theorem of Finitely Generated Abelian Groups

Question: The number of decomposition of \(G\) in the form \(\mathbb Z^r\times Z_{n_1}\dots\times Z_{n_s}\) is
Answer: exactly one.
Page: 158
Chapter: 5
Section: 2
Variables: \(G\) a group
Index: 3 (1)
Kind: Theo
Name: Fundamental Theorem of Finitely Generated Abelian Groups

Smaller0: \(H\) is the unique subgroup of its order
Greater0: \(H\) is characteristic
Page: 135
Chapter: 4
Variables: \(G\) a group
\(H\le G\).
Index: (2)

Name: \(H\) is a normal subgroup of \(G\)
Page: 082, 169,, 88, 89
Chapter: 3,5,, 3, 3
Section: 1, 5,,1, 1
Variables: \(G\) a group
\(H\subseteq G\).
Notation: \(H\trianglelefteq G\)
Index: 6, 7, 7, 26 (c), 37, 38, 39, 41
Kind: Theo, prop, Prop, Ex, Ex

Name: A normal subgroup of \(G\)
Definition: \(N_G(H)=G\)
Page: 082, 169,, 88, 89
Chapter: 3,5,, 3, 3
Section: 1, 5,,1, 1
Variables: \(G\) a group
Index: 6, 7, 7, 26 (c), 37, 38, 39, 41
Kind: Theo, prop, Prop, Ex, Ex
Definition2: \(H\) is the kernel of a morphism,
Example: \(A\times\{1_B\}\)
Conjdef: A subgroup \(H\le G\) such that, equivalently
Definition3: \(gN=Ng\) for all \(g\in G\),
Example2: \(\{(a,a)\mid a\in A\}\)
Definition4: the operation on left cosets of \(N\) in \(G\) is a group
Denoted by: \(N\)
French: Sous groupe distingué
Definition5: \(gSg^{-1}\subseteq H\) for \(S\) a generator of \(H\) and all \(g\in G\).
Example3: Commutator subgroup
Definition6: \([H,G]\le H\)
Applied to: \(A\times B\) for \(A,B\) groups
Counterexample: \(\{(a,a)\mid a\in S_3\}\) is not normal in \(S_3\times S_3\)
Applied to2: \(A\times A\) for \(A\) abelian

Question: The number of composition series of \(G\) is
Answer: exactly one, up to isomorphism of composition factors.
Page: 103
Chapter: 3
Section: 4
Variables: \(G\ne 1\) be a finite group.
Index: 22
Kind: Theo
Name: Jordan-Hölder

Name: The center of \(G\)
Definition: \(C_G(G)\).
Page: 050
Chapter: 2
Variables: \(G\) a group.
Notation: \(Z(G)\)
Kind: Def
Typ: A normal Abelian subgroup of \(G\)

Smaller0: \(|G/H|\)
Greater0: \(\frac{|G|}{|H|} \)
Page: 089
Chapter: 3
Section: 2
Variables: \(G\) a group and \(H\le G\).
Index: 8
Kind: Theo
Smaller to greater: =
Hide middle: yes
Name: Lagrange's Theorem

Question: The automorphism group of \(D_8\) is
Answer: \(D_8\).
Page: 136
Chapter: 4
Index: 17 (4)
Kind: Prop

Name: A presentation of a group
Definition: a set of generator \(S\)
Page: 026, 1
Chapter: 1,1 (generators and relations)
Section: 2,1
Definition2: a set of equalities \(R_i\) between terms of \(S^*\).
Conjdef: A pair containing
Name2: An abstract definition of a group
Denoted by: \(\langle S\mid R_1,\dots,R_m\rangle\)

Smaller0: the number of conjugacy classes of \(S_n\)
Greater0: the number of partition of \(n\)
Page: 126
Chapter: 4
Section: 3
Variables: \(n\in\mathbb N\).
Index: 11
Kind: Proposition
Smaller to greater: is equal to

Smaller0: abelian group
Greater0: the trivial action
Page: 123
Chapter: 4
Section: 3
Variables: \(G\) a group
\(a\in G\).
Prefix Greater: the action of \(G\) on itself by conjugation is
Greater1: \(\{a\}\)
Connect Greater0: and the conjugacy class of \(a\) is
Extra: counter example ?
Prefix Smaller: \(G\) is an

Name: The symmetric group of \(\Omega\)
Definition: The set of bijection on \(\Omega\).
Page: 029
Chapter: 1
Section: 3
Variables: \(\Omega\) a non-empty set with \(n\) elements
Notation: \(S_\Omega\)

Smaller0: prime
Greater0: \(G\) is cyclic
Page: 136, 90
Chapter: 4,3
Section: 4,2
Variables: \(G\) a group.
Index: ,10
Kind: Ex, Cor
Smaller1: \(pq\) with \(p<q\) primes and \(p\nmid q-1\)
Prefix Smaller: the order of \(G\) is
Connect Smaller0: or

Question: The order of \(S_\Omega\) is
Answer: \(n!=\Pi_{i=1}^n i\)
Page: 029
Chapter: 1
Section: 3
Variables: \(\Omega\) a non-empty set with \(n\) elements

Smaller0: the number of conjugates of \(S\) in \(G\) (\(\{gSg^{-1}\mid g\in G\}\))
Greater0: the index of the normalizer of \(S\) ([$]\abs{G:N_G(S)}[/$])
Page: 123
Chapter: 4
Section: 3
Variables: \(G\) a group
\(S\subseteq G\).
Index: 6
Kind: Proposition
Smaller to greater: is equal to
Construction: This is the special case of a more general result for a group action (conjugation).
The equivalent class of \(S\) is its number of conjugates. Its fixpoint is ith normalizer

Smaller0: \(A\le B\)
Greater0: \(A/N\le B/N\)
Page: 098
Chapter: 3
Variables: \(G\) be a group
\(N\trianglelefteq G\)
\(N\le A,B\le G\)
Index: 19
Kind: Theo
Smaller to greater: iff
Name: Fourth Isomorphism Theorem
Extra: The last one does not seems to appear in the theorem
Greatest: \(|B:A|=|B/N:A/N|\)

Smaller0: the product of two primes
Greater0: \(G\) is abelian
Page: 135
Chapter: 4
Variables: \(G\) a group.
Index: bottom
Kind: Ex
Prefix Smaller: the order of \(G\) is

Smaller0: \(pv=0\) for all \(v\in G\)
Greater0: an \(n\)-dimensional vector space over \(\mathbb F_p\)
Page: 136
Chapter: 4
Variables: \(p\) prime,
\(G\) an abelian additive groupe.
Index: 17 (3)
Kind: Prop
Hide middle: yes
Prefix Greater: \(G\) is
Smaller1: \(|G|=p^n\)

Question: The cardinal of \(F\) is
Answer: of the form \(p^n\) with \(p\) prime.
Page: 035
Chapter: 1
Section: 4
Variables: \((\mathbb F,+,\times)\) a finite field.

Name: The set generated by \(S\)
Definition: the set of elements of the form \(\pi_{i=0}^n s_i\) with \(s_i\) or \(-s_i\) belonging to \(S\) for all \(i\)
Page: 026
Chapter: 2
Section: 4
Variables: \(G\) a group
\(S\subseteq G\)
Notation: \(\langle S\rangle\)
Definition2: the intersection of subgroup of \(G\) containing \(S\).

Name: Cyclic sets
Page: 58, 84
Section: 3,1
Chapter: 2, 3
Index: 7, 2
Kind: Theo, ex
Under: taking a subgroup
Under2: taking quotient

Question: The automorphism group of \(\mathbb Z/n\mathbb Z\) is
Answer: isomorphic to \((\mathbb Z/n\mathbb Z)^\times\).
Page: 135
Chapter: 4
Variables: \(n\in\mathbb N\)
Index: 16
Kind: Prop

Name: Frobenius group of order 20, \(\langle(2354),(1 2 3 4 5)\rangle\)
Definition: Normalizer of Sylow 5-subgroup in \(S_5\)
Page: 168
Chapter: 5
Section: 3
Definition2: \(\langle x,y\mid x^4=y^5=1, xyx^{-1}=y^2\rangle\)
Extra: check:trouble with this page

Name: The commutator of \(x\) and \(y\)
Definition: \(x^{-1}y^{-1}xy\)
Page: 169
Chapter: 5
Section: 4
Variables: \(G\) a group
\(x,y\in G\)
Notation: \([x,y]\)
Intuition: Its value is 1 iff \(x\) and \(y\) commute

Name: The commutator of \(A\) and \(B\)
Definition: The subgroup generated by \(x^{-1}y^{-1}xy\) for \(x\in A\), \(y\in B\)
Page: 169
Chapter: 5
Section: 4
Variables: \(G\) a group
\(A,B\subseteq G\) non-empty
Notation: \([A,B]\)

Name: The commutator subgroup of \(G\)
Definition: \([G,G]\)
Page: 169
Chapter: 5
Section: 4
Variables: \(G\) a group
Notation: \(G'\)
Typ: characteristic in \(G\)

Question: \(yx[x,y]\)
Answer: equals \(xy\)
Page: 169
Chapter: 5
Section: 4
Variables: \(G\) a group
\(x,y\in G\)
\(H\le G\)
Index: 7
Kind: prop

Smaller0: \(\sigma[x,y]\)
Greater0: \([\sigma(x),\sigma(y)]\)
Page: 169
Chapter: 5
Section: 4
Variables: \(G\) a group
\(x,y\in G\)
\(H\le G\)
\(\sigma\) an automorphism of \(G\)
Index: 7(3)
Kind: prop
Smaller to greater: =

Smaller0: \(G/G'\)
Greater0: the largest Abelian quotient of \(G\)
Page: 169
Chapter: 5
Section: 4
Variables: \(G\) a group
\(x,y\in G\)
\(H\le G\)
\(\sigma\) an automorphism of \(G\)
Index: 7(4)
Kind: prop
Smaller to greater: =
Extra: construction ?

Smaller0: \(H\) Abelian
Greater0: \(G'\)
Page: 169
Chapter: 5
Section: 4
Variables: \(G,H\) groups
\(\phi:G\to H\)
Index: 7(5)
Kind: prop
Prefix Greater: \(\phi\) factors though

Question: \([r,s]=\)
Answer: \(r^{-2}\)
Page: 171
Chapter: 5
Section: 4
Variables: \(n\in\mathbb N\)
Context: \(D_{2n}\)
Index: 3
Kind: Ex

Smaller0: \(\sigma\) and \(\sigma'\) have the same type
Greater0: \(\sigma\) is a commutator iff \(\sigma'\) is a commutator
Page: 171
Chapter: 5
Section: 4
Variables: \(n\in\mathbb N\)
\(\sigma,\sigma'\) permutations in \(S_n\)
Kind: Ex

Smaller0: the number of way to write \(x\) as \(hk\)
Greater0: [$]\abs{H\cap K}[/$]
Page: 171
Chapter: 5
Section: 4
Variables: \(G\) a group
\(H,K\subseteq G\)
Index: 8
Kind: Prop
Smaller to greater: is equal to

Smaller0: \(H\), \(K\) normal in \(G\)
Greater0: \(HK\cong H\times K\)
Page: 171
Chapter: 5
Section: 4
Variables: \(G\) a group
\(H,K\subseteq G\)
\(x\in HK\)
Kind: Ex
Smaller1: \(H\cap K=1\)

Name: The internal direct product of \(H\) and \(K\)
Definition: generated by \(H\) and \(K\)
Page: 172
Chapter: 5
Section: 4
Variables: \(G\) a group
\(H,K\subseteq G\)
Notation: \(HK\)
Definition2: with elements of the form \(hk\), with \(h\in H\), \(k\in K\)
Conjdef: The subgroup of \(G\)
Assuming: \(H\cap K=1\), \(H,K\) normal

Name: The external direct product of \(H\) and \(K\)
Page: 172
Chapter: 5
Section: 4
Variables: \(G\) a group
\(H,K\subseteq G\)
Notation: \(H\times K\)

Definition: the greatest integer less than or equal to \(a\)
Page: 21
Chapter: 1
Variables: \(a\in\mathbb R\)
Notation: \([a]\)
Index: 7
Kind: ex

Name: The fractional part of \(a\)
Definition: \(a-[a]\)
Page: 21
Chapter: 1
Variables: \(a\in\mathbb R\)
Index: 7
Kind: ex

Name: The real numbers mod \(1\)
Definition: \((\mathbb R,+)/(\mathbb Z,+)\)
Page: 21
Chapter: 1
Index: 7
Kind: ex

Name: The roots of unity
Definition: The set of complex solutions of \(z^n=1\)
Page: 22
Chapter: 1
Variables: \(n\in\mathbb N\)
Index: 8
Kind: ex
Property: A group under multiplication

Question: [$]\abs{g^{-1}}[/$] is
Answer: [$]\abs{g}[/$]
Page: 22
Chapter: 1
Variables: \((G,\cdot)\) a group,
\(g\in G\)
Index: 20
Kind: ex

Smaller0: \(g^{n+n'}\)
Greater0: \(g^n\cdot g^{n'}\)
Page: 22
Chapter: 1
Variables: \((G,\cdot)\) a group,
\(g\in G\)
\(n,n'\in\mathbb Z\)
Index: 19
Kind: ex
Smaller to greater: =

Question: [$]\abs{g^{-1}hg}[/$]
Answer: equals [$]\abs{h}[/$]
Page: 22
Chapter: 1
Variables: \((G,\cdot)\) a group,
\(h,g\in G\)
\(n\in\mathbb N\)
Kind: ex

Smaller0: \(h,g\) commutes
Greater0: \((hg)^n\)
Page: 22
Chapter: 1
Variables: \((G,\cdot)\) a group,
\(h,g\in G\)
\(n\in\mathbb Z\)
Index: 24
Kind: ex
Greater1: \(h^ng^n\)
Connect Greater0: =

Smaller0: \(\prod_i G_i\) is Abelian
Greater0: each \(G_i\) is Abelian
Page: 23
Chapter: 1
Variables: \(I\) an indexing family
\((G_i,\cdot_i)_{i\in I}\) a group,
\(h,g\in G\)
\(n\in\mathbb N\)
Index: 29
Kind: ex
Smaller to greater: iff

Smaller0: \(x^i=x^j\)
Greater0: [$]i\equiv j\mod\abs{x}[/$]
Page: 23
Chapter: 1
Variables: \(G\) a group,
\(i,j\in\mathbb Z\)
\(g\in G\)
Index: 32
Kind: ex
Smaller to greater: iff

Question: the order of \(\sigma\)
Answer: is the lcm of the \(n_{i}\)
Page: 32
Chapter: 1
Section: 2
Variables: \(\sigma=(a_{1,1},\dots, a_{1,n_1})\dots(a_{m,1},\dots, a_{m,n_m})\) a permutation
Index: 15
Kind: Ex

Smaller0: the order of \(\sigma\) is \(p\)
Greater0: \(n_1=\dots=n_m=p\)
Page: 32
Chapter: 1
Section: 2
Variables: \(\sigma=(a_{1,1},\dots, a_{1,n_1})\dots(a_{m,1},\dots, a_{m,n_m})\) a permutation
\(p\) prime
Index: 13
Kind: Ex
Smaller to greater: iff

Question: The number of \(m\)-cycles in \(S_n\) is
Answer: \(\frac{n^{\underline m}}m\)
Page: 32
Chapter: 1
Section: 2

Question: The number of product of two disjoint cycles in \(S_n\) is
Answer: \(n^{\underline4}/8\)
Page: 32
Chapter: 1
Section: 2
Assuming: \(n\ge4\)

Name: The linear group of degree \(n\)
Definition: whose elements are \(n\times n\) matrices, with entry in \(\mathbb F\), whose determinant is non-zero
Page: 34
Chapter: 1
Section: 4
Variables: \(\mathbb F\) a field
\(n\in\mathbb N\)
Notation: \(GL_n(\mathbb F)\)
Definition2: whose operation is Matrix multiplication
Conjdef: The group

Smaller0: when each \(s_i\) are replaced by \(r_i\) in \(w_j=w_j'\), the equality holds
Greater0: there is a unique morphism \(G\to H\) sending \(s_i\) to \(r_i\)
Page: 38
Chapter: 1
Section: 6
Variables: \(G=\langle (s_i)_{i\in I}\mid (w_j=w_j')_{j\in J}\rangle\)
\(H\) a group
\((r_i)_{i\in I}\in H\)

Name: Abelian group
Page: 40
Section: 6
Chapter: 1
Index: 3
Kind: Ex
Under: morphism
Condition: surjective
Under2: product
Under3: quotient?

Name: The regular left action of \(G\)
Definition: The left action of \(G\) on \(G\) with [$]\defFun{G}{G\to G}{g}{\defFun{G}{G}{a}{ga}}[/$]
Page: 44
Chapter: 1
Section: 7
Variables: \(G\) a group

Name: A subgroup of \(G\)
Definition: \(H\) is a group
Page: 47
Chapter: 2
Section: 1
Variables: \(G\) a group
Kind: Prop
Definition2: \(xy^{-1}\in H\) for each \(x,y\in H\)
Conjdef: A non-empty subset \(H\) of \(G\) such that, equivalently

Name: A torsion element of \(G\)
Definition: whose order is finite
Page: 47
Chapter: 2
Section: 1
Variables: \(G\) a group
Index: 6
Kind: Ex
Definition2: which is a torsion element of \(G\) considered as a \(\mathbb Z\)-module (assuming \(G\) Abelian)
Conjdef: An element of \(G\), equivalently

Name: The torsion subgroup of \(G\)
Definition: containing all torsion element of \(G\)
Page: 47
Chapter: 2
Section: 1
Variables: \(G\) a group
Index: 6
Kind: Ex
Definition2: which is the Torsion submodule of \(G\), considered as a \(\mathbb Z\)-module
Conjdef: the subgroup of \(G\), equivalently
Assuming: \(G\) Abelian

Page: 47
Chapter: 2
Section: 1
Name: The special linear group
Set: The set of \(n\times n\)-matrices over \(F\) whose determinant is 1
Variables: \(F\) a field
\(n\in\mathbb N\)
Notation: \(SL_n(F)\)
*: Matrix multiplication
Kind: Ex
Index: 9
1: \(I_n\)
÷: Inverse

Name: The diagonal subgroup
Definition: The subgroup of \(G\times G\) containing the elements \((g,g)\) for each \(g\in G\)
Page: 49
Chapter: 2
Section: 1
Variables: \(G\) a group
Index: 11
Kind: Ex

Smaller0: \(H\) is Abelian
Greater0: \(H\le C_G(H)\)
Page: 52
Chapter: 2
Variables: \(G\) a group
\(H\le G\)
Index: 6b
Kind: Ex
Smaller to greater: iff

Question: \(H\le \mathbf{?}\le G\)
Answer: \(H\le N_G(H)\le G\)
Page: 52
Chapter: 2
Variables: \(G\) a group
\(H\le G\)
Index: 6b
Kind: Ex

Question: The number of cyclic group of cardinality \(n\) is
Answer: 1, \(\mathbb Z/n\mathbb Z\)
Page: 56
Chapter: 2
Section: 3
Variables: \(n\in\mathbb N\)
Index: 4
Kind: Theo

Question: The number of cyclic group of cardinality \(\infty\) is
Answer: 1, \(\mathbb Z\)
Page: 56
Chapter: 2
Section: 3
Variables: \(n\in\mathbb N\)
Index: 4
Kind: Theo

Smaller0: \(g,h\) commutes
Greater0: divides [$]lcm(\abs{g},\abs{h})[/$]
Page: 60
Chapter: 2
Section: 3
Variables: \(G\) a group
\(g,h\in G\)
Index: 16
Kind: Ex
Prefix Greater: [$]\abs{gh}[/$]
Counterexample: todoy

Smaller0: \(H\subseteq K\)
Greater0: \(\langle H\rangle\le\langle K\rangle\)
Page: 65
Chapter: 2
Section: 4
Variables: \(G\) a group
\(H,K\subseteq G\)
Kind: Ex
Smaller to greater: iff
Counterexample: \(\{1\}\) and \(\{-1\}\) in \(\mathbb Z\)

Smaller0: \(G\) is finitely generated
Greater0: a maximal element
Page: 66
Chapter: 2
Section: 4
Variables: \(G\) a group
Index: 17
Kind: Ex
Prefix Greater: the set of proper subgroup of \(G\) has
Assuming: Zorn's lemma

Smaller0: the \(G_i\)'s are divisible
Greater0: \(\prod G_i\) are divisible
Page: 66
Chapter: 2
Section: 4
Variables: \((G_i)_{i\in I}\) nontrivial Abelian groups
Index: 20
Kind: Ex
Smaller to greater: iff

Question: The lattice of subgroup of \(\mathbb Z/2\mathbb Z\) is
Page: 67
Chapter: 2
Section: 5
Index: 1
Kind: Ex

Question: The lattice of subgroup of \(\mathbb Z/4\mathbb Z\) is
Page: 67
Chapter: 2
Section: 5
Index: 1
Kind: Ex4

Question: The lattice of subgroups of \(\mathbb Z/p^n\mathbb Z\) is
Page: 67
Chapter: 2
Section: 5
Variables: \(p\) prime
\(n\in\mathbb N\)
Index: 1
Kind: Ex4

Question: The lattice of subgroups of \(\mathbb Z/6\mathbb Z\) is
Page: 68
Chapter: 2
Section: 5
Variables: \(p\) prime
\(n\in\mathbb N\)
Index: 1
Kind: Ex4

Question: The lattice of subgroups of \(\mathbb Z/12\mathbb Z\) is
Page: 68
Chapter: 2
Section: 5
Variables: \(p\) prime
\(n\in\mathbb N\)
Index: 1
Kind: Ex4

Question: The lattice of subgroups of \(V_4\) is
Page: 68
Chapter: 2
Section: 5
Index: 1
Kind: Ex4

Question: The lattice of subgroups of \(S_3\) is
Page: 69
Chapter: 2
Section: 5
Index: 1
Kind: Ex4

Question: The lattice of subgroups of \(D_8\) is
Page: 69
Chapter: 2
Section: 5
Index: 1
Kind: Ex4

Question: The lattice of subgroups of \(Q_8\) is
Page: 69
Chapter: 2
Section: 5
Index: 1
Kind: Ex4

Question: The lattice of subgroups of \(D_{16}\) is
Page: 70
Chapter: 2
Section: 5
Index: 6
Kind: Ex

Question: How to use a lattice of subgroups of \(G\) to compute centralizer of \(H\)
Answer: Restrict your study to elements between \(H\) and \(G\). Exclude those containing an element not commutating with an element of \(H\).
Page: 71
Chapter: 2
Section: 5
Variables: \(G\) a group
\(H\subseteq G\)

Question: The groups of order 4 are
Answer: \(Z_4\) and \(V_4\)
Page: 71
Chapter: 2
Section: 5
Index: 10
Kind: Ex

Name: Divisible Abelian groups
Page: 86
Section: 1
Chapter: 3
Kind: 15
Under: quotient
Condition: arbitrary
Under2: Product
Condition2: ?

Smaller0: non-empty
Greater0: the elements of order \(n\), for some arbitrary \(n\) fixed
Page: 88
Chapter: 3
Section: 1
Variables: \(G\) a group
Index: Ex, Ex
Kind: 15
Smaller to greater: contains
Hide middle: yes
Prefix Greater: the subgroup generated by
Greater to smaller: belongs to
Extra: 22, 23, 26(e)
Prefix Smaller: normal subgroups of \(G\) are:
-closed under
Connect Smaller0: intersection
-closed under
Suffix Smaller: join
-

Smaller0: \(a^gb^g\)
Greater0: \((ab)^g\)
Page: 88
Chapter: 3
Section: 1
Variables: \(G\) a group,
\(a,b,g\in G\)
\(\cdot^g\) being the conjugate of \(\cdot\)
Index: 26
Kind: Ex
Smaller to greater: =

Smaller0: \((a^{-1})^g\)
Greater0: \((a^g)^{-1}\)
Page: 88
Chapter: 3
Section: 1
Variables: \(G\) a group,
\(a,b,g\in G\)
\(\cdot^g\) being the conjugate of \(\cdot\)
Index: 26
Kind: Ex
Smaller to greater: =

Question: [$]\abs{a^g}[/$]
Answer: equals [$]\abs{a}[/$]
Page: 88
Chapter: 3
Section: 1
Variables: \(G\) a group,
\(a,b,g\in G\)
\(\cdot^g\) being the conjugate of \(\cdot\)
Index: 26
Kind: Ex

Question: \(N_G(N)\)
Answer: is the largest subgroup of \(G\) in which \(N\) is normal
Page: 88
Chapter: 3
Section: 1
Variables: \(G\) a group,
Index: 31
Kind: Ex

Smaller0: \(K\trianglelefteq H\)
Greater0: \(H\le N_G(K)\)
Page: 88
Chapter: 3
Section: 1
Variables: \(G\) a group,
\(K\le H\le G\)
Index: 31
Kind: Ex

Question: \(A\times B/A\times\{1_B\}\)
Answer: is isomorphic to \(B\)
Page: 89
Chapter: 3
Section: 1
Variables: \(A,B\) groups
Index: 37
Kind: Ex

Question: \(A\times A/\{(a,a)\mid a\in A\}\)
Answer: is isomorphic to \(A\)
Page: 89
Chapter: 3
Section: 1
Variables: \(A\) a group
Index: 37
Kind: Ex
Assuming: \(A\) Abelian

Smaller0: \(H,K\trianglelefteq G\)
Greater0: \(hk=kh\)
Page: 89
Chapter: 3
Section: 1
Variables: \(G\) a group
\(H,K\le G\)
\(h\in H\), \(k\in K\)
Index: 42
Kind: Ex
Smaller1: \(H\cap K=\{1_G\}\)

Question: The set of groups of order \(p\), for \(p\) prime, is
Answer: \(Z_p\)
Page: 90
Chapter: 2
Section: 3
Variables: \(p\in\mathbb N\)
Index: 10
Kind: Cor

Smaller0: \(G\) is Abelian
Greater0: non empty
Page: 92
Chapter: 3
Section: 2
Variables: \(G\) a finite group
\(n\) an integer
Prefix Greater: the set of subgroups of \(G\) of order \(n\) is
Smaller1: \(n\) a dividor of [$]\abs{G}[/$]

Question: the order of \(H_1\cap H_2\)
Answer: divides the gcd of the order of the \(H_i\)
Page: 95
Chapter: 3
Section: 2
Variables: \(G\) a group
\(H_1,H_2\le G\)

Smaller0: the two index [$]\abs{G:H_i}[/$] are relatively prime
Greater0: equals [$]\abs{G:H_1}\abs{G:H_2}[/$]
Page: 95
Chapter: 3
Section: 2
Variables: \(G\) a group
\(H_1,H_2\le G\)
Prefix Greater: [$]\abs{H:H_1\cap H_2}[/$]

Question: The stabilizer of \(i\) in \(S_n\) is
Answer: isomorphic to \(S_{n-1}\)
Page: 95
Chapter: 3
Section: 2
Variables: \(0<1\le n\)
Index: 15
Kind: ex

Question: Example of group with no proper subgroup of finite index
Answer: Abelian divisible groups
Page: 95
Chapter: 3
Section: 2
Index: 21
Kind: ex

Question: How to obtain the lattice of \(G/N\)
Answer: Take the part of the lattice of \(G\), above \(N\). Replace in it each group \(H\) by \(H/N\)
Page: 99
Chapter: 3
Variables: \(G\) a group
\(N\trianglelefteq G\)

Question: [$]\abs{GL_n(F):SL_n(F)}[/$] equals
Answer: \(\mathrm{char}(F)-1\)
Page: 101
Chapter: 3
Variables: \(F\) a field
\(n\in\mathbb N\)
Index: 1
Kind: Ex

Smaller0: [$]\abs{G:H}[/$] prime
Greater0: =\(G\)
Page: 101
Chapter: 3
Variables: \(G\) a group
\(H,K\le G\)
Kind: ex
Prefix Greater: (\(HK\)
Greater1: =[$]\abs{G:H}[/$]
Connect Greater0: and [$]\abs{K:K\cap H}[/$]
Greater2: \(K\le H\)
Connect Greater1: ) or

Name: The relation \(H\trianglelefteq G\) (being a normal subgroup)
Page: 101
Section: 3
Index: 4
Kind: Ex
Under: cartesian product of the sub and super group
Condition: arbitrary

Page: 66
Chapter: 2
Section: 4
Name: The group of \(p\)-power roots of \(1\)
Set: Complex number \(z\) such that \(z^{p^n}=1\) for some \(n\in\mathbb Z\)
Variables: \(p\) prime
*: the multiplication
Kind: Ex
Index: 18
÷: \(z^{p^n}\mapsto z^{-p^n}\)
Base: Not finitely generated

Page: 66
Chapter: 2
Section: 4
Name: The group of \(p^k\)-th roots of unity
Set: Complex number \(z\) such that \(z^{p^k}=1\)
Variables: \(p\) prime
\(k\in\mathbb N\)
*: with multiplication
Kind: Ex
Index: 18
1: 1
÷: \(z^{p^k}\mapsto z^{-p^k}\)
Typ: The group

Smaller0: the group of \(p^k\)-th roots of unity
Greater0: the group of \(p^{k+1}\)-th roots of unity
Page: 66
Chapter: 2
Section: 4
Variables: \(p\) prime
\(k\in\mathbb N\)
Index: 18
Kind: Ex
Smaller to greater: is a subgroup of
Greater to smaller: is a supergroup of
Hide sides: yes

Question: The proper subgroups of the group of \(p\)-power roots of unity
Answer: are exactly the groups of \(p^k\)-th roots of unity
Page: 66
Chapter: 2
Section: 4
Variables: \(p\) prime
Index: 18
Kind: Ex

Smaller0: \(H\) Hall subgroup
Greater0: \(H\cap K\) is a Hall subgroup of \(N\)
Page: 101
Chapter: 3
Variables: \(G\) a group
\(H,K\le G\)
Index: 10
Kind: Ex
Smaller1: \(K\trianglelefteq G\)
Greater1: \(HN/N\) is a Hall subgroup of \(G/N\)

Smaller0: \(n=2\)
Greater0: \(SL_n(\mathbb F_p)\) is simple
Page: 104
Chapter: 3
Variables: \(p\) prime
\(n\in\mathbb N\)
Smaller to greater: or
Smaller1: \(p\) is 2 or 3

Smaller0: \(G\) is Abelian
Greater0: \(G\) is isomorphic to \(Z_p\) for some prime \(p\)
Page: 106
Chapter: 3
Section: 4
Variables: \(G\) a group
Smaller1: \(G\) is simple

Name: Solvable groups
Page: 106
Section: 4
Chapter: 3
Index: 5
Variables: \(G\) a group
Kind: ex
Under: taking a subgroup
Condition: arbitrary
Under2: quotient by a subgroup
Condition2: an arbitrary normal

Smaller0: \(H\trianglelefteq G\)
Greater0: \(H\) occurs in a composition series of \(G\)
Page: 106
Chapter: 3
Section: 4
Variables: \(G\) a group
\(H\le G\)
Index: 7
Kind: ex
Smaller1: \(G\) finite

Smaller0: \(H\trianglelefteq G\)
Greater0: \(H\) occurs in a composition series of \(G\)
Page: 106
Chapter: 3
Section: 4
Variables: \(G\) a group
\(H\le G\)
Index: 7
Kind: ex
Smaller1: \(G\) finite
Construction: Let \(1=H_0\triangleleft \dots \triangleleft H_n=H\), \(1=\overline G_1\triangleleft\dots\triangleleft \overline G_m=G/H\), chains of normal subgroups whose pairwise quotients are Abelian.
By induction on \(i\) from \(m\) to \(0\), by lattice isomorphism theorem, there exists \(H\le G_i\trianglelefteq G_{i+1}\) such that \(G_i/H=\overline G_i\).
By third isomorphism theorem, \(\overline{G_{i+1}}/\overline {G_i}=(G_{i+1}/H)/(G_i/H)\cong G_{i+1}/G_i\).
Finally, take \(1=H_0\triangleleft \dots \triangleleft H_n=H=G_0\triangleleft \dots \triangleleft G_m=G\)

Smaller0: \(G/H\) is simple
Greater0: there exists \(H\triangleleft I\triangleleft G\)
Page: 106
Chapter: 3
Section: 4
Variables: \(G\) a group
\(H\le G\)
Index: 7
Kind: ex
Smaller to greater: or
Assuming: \(H\trianglelefteq G\)

Name: normal subgroup relation
Section: 1
Chapter: 3
Closure: not
Under: transitivity
CounterExample: \(\langle s\rangle\trianglelefteq\langle s,r^2\rangle \trianglelefteq D_8\).

Smaller0: \(Z(G)\)
Greater0: \(G'\)
Page: 171
Chapter: 5
Section: 4
Variables: \(G\) a group
Index: 2
Kind: Ex
Smaller to greater: \(\subseteq\)
Greater to smaller: \(\supseteq\)

Smaller0: \(K\trianglelefteq G\)
Greater0: \(H\cap K\trianglelefteq H\)
Page: 86
Chapter: 3
Section: 1
Variables: \(G\) a group
\(H,K\le G\)
Index: 15
Kind: Ex
Extra: 24

Name: A generator
Definition: An element \(g\in G\) such that each \(h\in G\) is of the form \(g^z\) with \(z\in \mathbb Z\)
Page: 54
Chapter: 2
Section: 3
Variables: \(G\) a group
Context: Element of \(G\)

Name: A generator
Definition: each element of \(G\) a product of a finite number of elements of \(H\), and of their inverse, possibly with repetition.
Page: 54
Chapter: 2
Section: 3
Variables: \(G\) a group
Context: Subset of \(G\)
Definition2: \(G\) is the group generated by \(H\)
DefType: A subset \(H\subseteq G\) such that, equivalently

Name: perfect field
Under: union
Condition: the set of fields is totally ordered by extension
Under2: fields algebraic over a perfect field

Question: The set of left coset of \(H\) in \(G\) is
Answer: a partition of \(G\)
Page: 80
Chapter: 3
Section: 1
Variables: \(G\) a group
\(H\) a subgroup of \(G\)

Name: \(Hg\)
Page: 76
Section: 1
Chapter: 3
Variables: \(G\) a group
\(H\) a subgroup of \(G\)
\(Hg\) a right coset of \(H\) in \(G\)
Closure: left multiplication by an element of \(H\)

Name: the set of left cosest of \(H\) in \(G\)
Page: 77
Section: 1
Chapter: 3
Variables: \(G\) a group
\(H\) a subgroup of \(G\)
Under: multiplication of left cosets
Under2: left multiplication by an element of \(G\)

Question: The pointwise inverse of a left coset of \(H\) in \(G\) is
Answer: a right coset of \(H\) in \(G\)
Page: 95
Chapter: 3
Section: 1
Variables: \(G\) a group
\(H\) a subgroup of \(G\)
Index: 12
Kind: ex