Definitions of Equivalence & Groups

(1) State the definition of equivalence relation.... and (2) Give one example of an abelian group and two (2) examples of nonabelian groups

Group Homomorphism

Let phi: G ---> H be a group homomorphism. Show that phi[G] is abelian if and only if for all x, y in G, we have xyx^(-1)y^(-1) in ker(phi). proving (=>) seems almost obvious since if it is abelian that means xyx^(-1)y^(-1) = xx^(-1)yy^(-1)=ee which is in the kernel, but I'm not sure about how to do the reverse (<=) and show that phi is abelian?

Symmetric Group Question

It is trivial that S[n] is cyclic for n = 1, 2, but is S[n] ever cyclic for n>=3? Prove why or why not.

Normal Subgroups

Find all maximal normal subgroups of Z[p] × Z[q], where p and q are relatively prime. Would the elements from Z[p] have to be one that are relatively prime to q and vice versa?

Group of Permutations

Consider the group Z[4] × Z[6] under * such that (a, b) * (c, d) = (a +[4] c, b +[6] d). (here +[4] means + is in Z[4] and +[6] is in Z[6]) We would like to find a group of permutations that is isomorphic to Z[4]Z[6]. Is this group cyclic? If so, prove it. If not, explain why. Do I need to list all the members and check or is it enough to know that Z[2]xZ[12] has the same order. And then check that their identity elements have the same order?