3. Let G = {u, v, w, x, y, z} be the 6-element group that has been assigned to you.

(i) Find the identity element e of G.

(ii) Find the inverse of each element of G.

(iii) For each element of G, find the cyclic subgroup that it generates.

(iv) Find all the subgroups of G.

(v) Choose an element of order 3 in G, and call it a. Find all the distinct cosets of H = (a) in G, and write out the multiplication table for the quotient group G/H.

(vi) Choose an element of order 2 in G, and call it b. Let Z5 = {0, 1, 2, 3, 4} be the 5-element cyclic group under addition modulo 5. Let (K, ∗) be the direct product of Z5 and (b).

Construct the Cayley table for (K, ∗).