Group
A group is a non-empty set $G$ together with a binary operator $\cdot: G \to G$. The following three requirements called the group axioms have to be met:
- $\forall a, b, c \in G, (a \cdot b) \cdot c = a \cdot (b \cdot c).$
- $\exists e \in G : \forall a \in G, e \cdot a = a \cdot e = a.$
- $\forall a \in G, \exists b \in G : a \dot b = e.$
Remarks
Additionally, a group is a:
- Monoid with invertibility.