[[Group theory MOC]]
# Group
A **group** is a [[Monoid]] with the additional requirement that every element have an inverse.
A group need not be commutative; this is a special case known as an [[Abelian group]]. #m/def/group

4. **Inverse** for each $a \in G$ there exists (provably unique) $a^{-1} \in G$ such that $a^{-1}a = e = aa^{-1}$

Groups play an important role in describing [[Symmetry]].
The concept of a group may be generalised to the concept of a [[Groupoid]],
which can be thought of as a typed group.

## Terminology and notation
Typically, given a group $G$ the identity element is denoted $e$ (for _Einheit_).
Usually multiplicative notation is used so that juxtaposition or $\cdot$ is the group operation and $a^n$ represents repeated operation.
For some [[Abelian group|abelian]] groups addition notation may be used where $+$ is the group operation and $na$ represents repeated operation.

- Both groups and group elements can be assigned [[Group order|order]].
- A subset of a group that remains closed under the operation is a [[Subgroup]]. 
  $\{ e \}$ is the trivial subgroup.



## Examples
See [[Examples of groups]]

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