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Problem session exercises 1. Let G be a cyclic group, with identity element e. Then G = hai for some a ∈ G. Let H be a subgroup of G. Prove: (a) The set S = {k ∈ N : ak ∈ H} is either empty, or contains a least element. (b) If S = ∅ then H = hei. (c) Now suppose that S 6= ∅, and let m be the least element of S. Then ham i ⊆ H. (d) If an ∈ H and n = mq + r where r, q ∈ Z, then ar ∈ H. (e) If an ∈ H and n = mq + r where r, q ∈ Z and 0 ≤ r < m, then r = 0. (f) H = ham i. (g) Every subgroup of a cyclic group is a cyclic subgroup. (h) The subgroups of (Z, +) are precisely the sets kZ = {kn : n ∈ Z} for k ∈ Z.