(1) Each of the following is incorrect. Explain briefly why or give a counterexample. (a) Addition is a binary operation on the set S = {-1,0, 1}. (b) If x, y and z are elements of a group G and xy =...

Extracted text: (1) Each of the following is incorrect. Explain briefly why or give a counterexample. (a) Addition is a binary operation on the set S = {-1,0, 1}. (b) If x, y and z are elements of a group G and xy = zx, then y = z. (c) If a is a nonidentity element of a group G and m is a positive integer such that a" = e, then la = m. (d) A cyclic group G of order 8 has a subgroup of order 4 that is isomorphic to the Klein-4 group. (e) Every nonidentity element of a cyclic group G generates G.