Consider a nonempty set G with a binary operation @ such that for elements a and b in the group, the element a@b is also in G. G is a group if the following properties hold.

  • The binary operation @ is associative. That is to say, for all elements a, b, c in G, it holds that a@(b@c) = (a@b)@c.
  • There exists an identity element e such that a@e = e@a = a for all elements a in G.
  • Every element a of G has an inverse, a-1 in G such that a@a-1 = a-1@a = e.