Let ø: G1 → G2 be a group homomorphism. (a) Suppose H is a subgroup of G1. Define ø(H)= {¢(h) | h E H}. Prove that ø(H) is a subgroup of G2. (b) Let ker(ø) = {g € G1 | ø(g) = e2} where e2 is the identity in G2. Prove that ker(ø) is a subgroup of G1. (c) Prove that ø is a group isomorphism if and only if ker(ø) = {e1} where e¡ is the identity of G1 and ø(G1) = G2.

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