Section
Algebra: Groups
Group theory through Sylow theorems and structure
Core idea
This section contains definitions, theorems, lemmas, propositions, and corollaries from group theory.
Definitions
Basic Structures
Subgroups
- Subgroup
- Trivial subgroup
- Proper subgroup
- Cyclic subgroup
- Generated subgroup
- Normal subgroup
- Characteristic subgroup
Special Groups
Centralizers, Normalizers, and Centers
Conjugacy
Commutators and Derived Series
- Commutator of elements
- Commutator subgroup (derived subgroup)
- Derived series
- Lower central series
- Upper central series
p-Groups and Sylow Theory
Series and Composition
Homomorphisms
Cosets and Quotients
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
Presentations and Free Groups
Extensions and Exact Sequences
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Theorems
Isomorphism Theorems
- First isomorphism theorem
- Second isomorphism theorem
- Third isomorphism theorem
- Correspondence theorem
Fundamental Theorems
Actions and Counting
Sylow Theorems
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
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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
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Propositions
Basic Properties
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
Conjugation and Order
Cyclic Groups
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
Extensions
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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
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Uncategorized
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