3. For each of the following groups G and subsets HCG, determine if H is a subgroup of G and prove your answers. (a) G=C, H = {a+bi | a, b = Z}. (b) G=C, H = {z € C | |z| = 1}. (c) G=C, H = {a + bi | a, b = Z, (a, b) = (0,0)}.

1. Consider the group U(5).(a) What is |U(5)|?(b) For each a ∈ U(5), find |a|, < a >, and | < a > |   2. Consider the group Z6.(a) What is Z6?(b) For each a ∈ Z6, find |a|, < a >, and | < a > |.

Determine whether or not H is a subgroup of G.

4*. Let f G H be a group homomorphism. Prove: (a) If S G then f(S) 4 f(G) (b) Show by example that S aG need not imply f(S) (c) If T H then f1(T) G. H