If G is cyclic prove that GH must also be cyclicSolutionProo

If G is cyclic, prove that G/H must also be cyclic

Solution

Proof. Suppose that G is cyclic, and let a G be a generator for G. We claim that the coset Ha generates G/H. It suffices to show that if Hb G/H is any right coset of H, then we have

Hb = (Ha)^m for some m Z.

Well, b G = <a>, so there exists an integer m such that b = a^m. Then we have

(Ha)^m = (Ha)(Ha)· · ·(Ha) = Ha^m = Hb. Since Hb was arbitrary,

we see that G/H = (Ha), so the quotient group is cyclic.