# different definitions of a group

A group is classically defined as a set with a binary operation (the group product) which is associative, such that there is a unit and for every element there is an inverse.

I know we can define a group in many different ways: for example as a heap, that is a set with a ternary operation $[x,y,z]$ satisfying certain properties.

Do you know other (equivalent) definitions of group? Also any reference to textbooks/papers/other accounting this problem will be accepted as an answer.

• Here is an alternative definition : planetmath.org/alternativedefinitionofgroup But it is not that different from the usual definition. – Mathematician 42 Apr 7 '16 at 9:34
• Basically there is only one definition of a group. More interesting are different definitions of, say, solvable groups - see this MSE-question. – Dietrich Burde Apr 7 '16 at 9:41
• Would something like "a group is a monoid with both a left and a right inverse" interest you? Or "a finite group is a finite cancellative semigroup". (Neither are awfully difficult to prove.) – user1729 Apr 7 '16 at 11:20