To determine whether a set is a group we need to check the following axioms:
1) Closure under the operation;
2) Associativity of the operation, that is, for any a,b,c in the set (a.b).c = a.(b.c) (here . is the operation)
3) Existence of the identity element e, such that for any a a.e = e.a = a
4) Existence of the inverses, that is for any a there is ...
Three examples of sets each equipped with a binary operation checked for being a group with respect to that operation, by checking the axioms directly.