Algebra: Groups

This section contains definitions, theorems, lemmas, propositions, and corollaries from group theory.

Definitions

Basic Structures

• Semigroup
• Monoid
• Group
• Abelian group

Subgroups

• Subgroup
• Trivial subgroup
• Proper subgroup
• Cyclic subgroup
• Generated subgroup
• Normal subgroup
• Characteristic subgroup

Special Groups

• Simple group
• Solvable group
• Nilpotent group
• Perfect group

Centralizers, Normalizers, and Centers

• Center of a group
• Centralizer
• Normalizer

Conjugacy

• Conjugate element
• Conjugacy class
• Class function

Commutators and Derived Series

• Commutator of elements
• Commutator subgroup (derived subgroup)
• Derived series
• Lower central series
• Upper central series

p-Groups and Sylow Theory

• p-group
• Sylow p-subgroup
• Hall subgroup

Series and Composition

• Composition series
• Subnormal series
• Chief series

Homomorphisms

• Group homomorphism
• Group monomorphism
• Group epimorphism
• Group isomorphism
• Kernel
• Image

Cosets and Quotients

• Coset (left/right)
• Index of a subgroup
• Quotient group

Products

• Direct product of groups
• Direct sum of groups
• Internal direct product
• Semidirect product
• Internal semidirect product

Group Actions

• Group action
• Orbit
• Stabilizer
• Fixed-point set
• Kernel of an action
• Faithful action
• Free action
• Transitive action
• Regular action
• Permutation representation
• Conjugation action

Automorphisms

• Automorphism group
• Inner automorphism
• Outer automorphism group

Presentations and Free Groups

• Group presentation
• Generating set
• Free group
• Normal closure

Extensions and Exact Sequences

• Group extension
• Split extension
• Central extension
• Exact sequence of groups

Theorems

Isomorphism Theorems

• First isomorphism theorem
• Second isomorphism theorem
• Third isomorphism theorem
• Correspondence theorem

Fundamental Theorems

• Cayley's theorem
• Lagrange's theorem
• Cauchy's theorem (finite groups)

Actions and Counting

• Orbit-stabilizer theorem
• Class equation
• Burnside's lemma

Sylow Theorems

• Sylow's first theorem
• Sylow's second theorem
• Sylow's third theorem

Structure Theorems

• Jordan-Hölder theorem
• Schreier refinement theorem
• Fundamental theorem of finitely generated abelian groups

Advanced Theorems

• Nielsen-Schreier theorem
• Schur-Zassenhaus theorem
• Burnside's p^a q^b theorem
• Krull-Remak-Schmidt theorem

Lemmas

• Subgroup test (one-step)
• Subgroup test (two-step)
• Normal subgroup criterion
• Subgroup of index 2 is normal
• p-group has nontrivial center
• Orbit decomposition lemma
• Conjugacy class size lemma
• Sylow conjugacy lemma
• Frattini argument
• Schreier's lemma
• Cosets partition a group
• Universal property of quotient groups
• Kernels are normal subgroups

Propositions

Basic Properties

• Uniqueness of identity
• Uniqueness of inverses
• Cancellation laws

Subgroup Properties

• Subgroups closed under inverses and products
• Intersection of subgroups is a subgroup
• Product of normal subgroups is normal
• Center is characteristic

Homomorphism Properties

• Kernel is normal
• Image is a subgroup
• G/ker(f) ≅ im(f)

Conjugation and Order

• Conjugation preserves order

Cyclic Groups

• Subgroups of cyclic groups are cyclic
• Finite cyclic group ≅ ℤ/nℤ
• Aut(cyclic of order n) ≅ (ℤ/nℤ)×

Actions

• Group acts on itself by left multiplication
• Group acts on itself by conjugation
• Class equation decomposition

Order and Structure

• |G| prime implies G cyclic
• |G| = p² implies G abelian
• Abelian implies all subgroups normal
• Finite p-group has subgroups of every order p^k

Sylow Properties

• n_p = 1 implies Sylow p-subgroup is normal

Extensions

• Semidirect product from splitting exact sequence

Corollaries

• Fermat's little theorem
• Euler's theorem
• Order of element divides order of group
• Finite p-group has nontrivial center
• n_p ≡ 1 mod p
• Classification of finite abelian groups
• Jordan-Hölder uniqueness