[[Abstract algebra MOC]]
# Group theory MOC

**Group theory** is predominantly concerned with symmetry, and its algebraic representation as a [[group]], the fundamental object of study.

## Objects

- [[Group]]

### Type of group


- [[Torsion group]]
- [[p-group]]
- [[Nilpotent group]]
- [[Solvable group]]
- [[Abelian group]]
  - [[Elementary abelian group]]
- [[Finite group]]
  - [[Cyclic subgroup|Cyclic group]]
- [[Simple group]]
- [[Topological group]]
  - [[Discrete group]]
  - [[Compact group]]
  - [[Lie group]]


### Special groups

- [[Symmetric group]]
- [[Alternating group]]
- [[General linear group]]

## Morphisms

- [[Group homomorphism]]
  - [[Kernel of a group homomorphism]]
 - [[Group action]]
   - [[Group representation theory MOC]]

## Internally

- [[Conjugation by an element]]
  - [[Group class function]]
- [[Coset]]
- [[Torsion group]]
- [[Subgroup]] (subobjects)
  - [[Cyclic subgroup]]
  - [[Normal subgroup]]
  - [[Core of a subgroup]]

### Commutation

- [[Centre of a group]]
- [[Centralizer in a group]]
- [[Group commutator]]


## Externally

- [[Quotient group]] (quotient objects)
- [[Direct product of groups]] (product) [[Free product of groups]] (coproduct)
- [[Semidirect product]]

### Functors in

- [[Free group]]
  - [[Group presentation]]



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