A group is a set G together with a binary operation ⋅:G×G→G such that:
- (Associativity) For all a,b,c∈G, (a⋅b)⋅c=a⋅(b⋅c).
- (Identity) There exists an element e∈G such that for all a∈G, e⋅a=a and a⋅e=a.
- (Inverses) For every a∈G there exists an element a−1∈G such that a⋅a−1=e and a−1⋅a=e.
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.