Group

A group is a mathematical structure that abstracts the behavior of symmetries acting on an object. It consists of a set with a given binary operation on it satisfying certain axioms.

Groups are extremely useful throughout mathematics, and are central to the theory of polytopes. Lots of research on them has been devoted to studying their most symmetric classes, including regular and uniform polytopes. The ways in which these are symmetric may be described with groups, including symmetry groups and automorphism groups.

Definition
A group is a set $$G$$ together with a binary operation $$*:S\times S\to S$$ (that is, the operation is closed) such that:


 * For any $$a,b,c\in S$$, $$(a*b)*c=a*(b*c)$$. (Associativity)
 * There exists $$e\in S$$ such that $$a*e=e*a=a$$ for any $$a\in S$$. (Identity element)
 * For any $$a\in S$$ there exists $$a^{-1}\in S$$ such that $$a*a^{-1}=e$$.