1. The problem statement, all variables and given/known data (i) Every group-theoretic relation p=q satisfied by (a,b,c) in G a group is also satisfied by (x,y,z) in F a group. (ii) There exists a homomorphism between G and F a->x b->y c->z. Problem: Show by example (i) can hold and (ii) cannot. Show (i) can hold and (ii) can hold but not be a unique homomorphism. 3. The attempt at a solution I'm having trouble trying to solve the first part of this question. Any help would be appreciated. As for part 2. Let G = (ZxZxZxZ, +) e_i is the identity in all spots except ith spot of the 4-tuple, 1 otherwise. EX: e_1 = (1,0,0,0) a = e_1 b = e_2 c = e_3 Its pretty clear that any relation p satisfied by a,b,c must be equal to the form: A_1(a) + A_2(b) + A_3(c) where A_i is the # of times you add or subtract. Let F = (ZxZxZxZ, +) x = e_1 b = e_2 c = e_3 e_4 -> e_4 Obviously this is one homomorphism were (i) and (ii) both hold x = e_1 b = e_2 c = e_4 e_4 -> e_3 This is another homomorphism were (i) and (ii) both hold. Thus the homomorphism is not unique.