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.