Let S be a nonempty finite set with a binary operation * that satisfies the associative law. Show that S is a group if a * b = a * c implies b = c and a *c = b * c implies a = b for all a, b, c E S. What can you say if S is infinite?

Let's consider a group G with the operation * defined as follows: For any two elements a, b in G, a * b = 2a + 3b. Given that G = {1, 2, 3, 4, 5} and the operation * is associative, find the identity element e in the group G.

For each binary operation * defined on a set below, determine whether or not * gives a group structure on the set. If it is a group, explain why it is a group. If it is not a group, say which axioms fail to hold: (a) Define * on Z by a * b max(a, b). (b) Define * on Q \ {0} by a * b = |ab|. (c) Define * on Q+ by a * b = ab.

Determine whether the set G is a group under the operation * G={n integer|n is odd}; a*b=a+b