Index

transposition

Transpositions

1 step subgroup test

Exercises for Chapter 2

1-1

Sets and mappings

$Aut(G)$

Special Homomorphisms and Isomorphisms

$Inn(G)$

Special Homomorphisms and Isomorphisms

$gcd$

Number theory

$p$-group

The second and third

$lcm$

Number theory

$\phi(m)$

Number theory

${\cal R} (m)$

Number theory

$\mathbb{Z}$

Definition of a group

$k$-cycle

Exercises

$C_G(a)$

Conjugacy

$Cl(a)$

Conjugacy

$[G: H]$

Cosets and Lagrange's Theorem

$\mathbb{Z}_m$

Exercises

$gp(S)$

Generators and isomophisms

$\mu_n$

Generators and isomophisms

$N_G(H)$

Normal subgroups

$\mathbb{Z}/m\mathbb{Z}$

Exercises

abelian

Definition of a group

alternating group

Transpositions

associative law

Definition of a group

automorphism

Special Homomorphisms and Isomorphisms

binary operation

Definition of a group

cartesian product

Sets and mappings

Cauchy's Theorem

Existence of Sylow subgroups;

Cayley table

Definition of a group

center

Exercises for Chapter 2

centralizer

Exercises for Chapter 2 | Conjugacy

class equation

Cosets and Lagrange's Theorem

closed

Definition of a group

co-domain

Sets and mappings

commutative

Definition of a group

commutator subgroup

Commutators and solvable groups

complete residue system

Number theory

complex

Subsets of a group

complexes $\mathbb{C}$

Definition of a group

composition series

The Jordan-Hölder Theorem

conjugacy class

Conjugacy

conjugate subgroups

Normal subgroups

coprime

Number theory

Correspondence Theorem

Isomorphism theorems

cosets, left

Cosets and Lagrange's Theorem

cycle

Cycles and cycle notation

cyclic

Generators and isomophisms

cyclic group

Generators and isomophisms

cyclic group $\mu_n$

Definition of a group

cyclic permutation

Cycles and cycle notation

derived series

Exercises

derived subgroup

Commutators and solvable groups

direct product

External and internal direct | External and internal direct

disjoint cycles

Cycles and cycle notation

disjoint sets

Sets and mappings

divides

Number theory

Division Algorithm

Number theory

domain

Sets and mappings

double coset

Double cosets

empty set $\emptyset$

Sets and mappings

endomorphism

Special Homomorphisms and Isomorphisms

equivalence class

Sets and mappings

equivalence relation

Sets and mappings

equivalent (series)

Series of groups; solvable

Euclidean Algorithm

Number theory

Euler $\phi$-function

Number theory

Euler's theorem

Cosets and Lagrange's Theorem

even permutation

Transpositions

external direct product

External and internal direct

factor group

Factor groups

factors

Series of groups; solvable

Fermat's Little Theorem

Cosets and Lagrange's Theorem

Finite Subgroup Test

Exercises for Chapter 2

first isomorphism theorem

Isomorphism theorems

first Sylow Theorem

Existence of Sylow subgroups;

function

Sets and mappings

generated by, group

Generators and isomophisms

generator

Generators and isomophisms

GL(n)

Definition of a group

greatest common divisor

Number theory

group

Definition of a group

homomorphism

Definition and Elementary Properties

identity element

Definition of a group

identity map

Sets and mappings

image

Sets and mappings

image set

Sets and mappings

index

Cosets and Lagrange's Theorem

infinite order

Elementary Properties of Groups

injective

Sets and mappings

internal direct product

External and internal direct

intersection

Sets and mappings

inverse mapping

Sets and mappings

isomorphic

Generators and isomophisms

isomorphic (series)

Series of groups; solvable

isomorphism

Generators and isomophisms

Jordan-Hölder theorem

The Jordan-Hölder Theorem

Klein 4-group

Exercises | Normal subgroups

Lagrange's Theorem

Cosets and Lagrange's Theorem

least common multiple

Number theory

left cosets

Cosets and Lagrange's Theorem

mapping

Sets and mappings

maximal (normal subgroup)

The Jordan-Hölder Theorem

multiplication table

Definition of a group

normal series

Series of groups; solvable

normal subgroup

Normal subgroups

normalizer

Normal subgroups

odd permutation

Transpositions

one-to-one

Sets and mappings

onto

Sets and mappings

order $o(g)$

Elementary Properties of Groups

order of a set

Sets and mappings

partition

Conjugacy

partitioned

Sets and mappings

permutation

Definition of a group

positive integers $\mathbb{Z}_+$

Definition of a group

pre-image

Sets and mappings

prime

Number theory

primitive $n^{th}$ root of unity

Generators and isomophisms

quotient group

Factor groups

rationals $\mathbb{Q}$

Definition of a group

refinement of a series

Series of groups; solvable

relatively prime

Number theory

residue classes

Number theory

Schreier's theorem

Series of groups; solvable

second fundamental isomorphism theorem

Isomorphism theorems

second Sylow Theorem

The second and third

semi-group

Definition of a group

series (of groups)

Series of groups; solvable

simple

Simple groups

SL(n)

Elementary Properties of Groups

special linear group

Elementary Properties of Groups

subset

Sets and mappings

Sylow Theorem

Existence of Sylow subgroups; | The second and third | The second and third

symmetric group $S_n$

Definition of a group

third isomorphism theorem

The third isomorphism theorem

third Sylow Theorem

The second and third

torsion free

Elementary Properties of Groups

union

Sets and mappings

Wardlaw

Cyclic Groups

Wardlaw, W.

Exercises