Give a one-sentence synopsis of the proof of Theorem 15.9.

Data from Theorem 15.9.

A factor group of a cyclic group is cyclic. 

Proof Let G be cyclic with generator a, and let N be a normal subgroup of G. We claim the coset aN generates G/N. We must compute all powers of aN. But this amounts to computing, in G, all powers of the representative a and all these powers give all elements in G. Hence the powers of aN certainly give all cosets of N and G/N is cyclic.