Give at most a three-sentence synopsis of the proof of Theorem 6.6.

6.6 Theorem A subgroup of a cyclic group is cyclic. 

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Proof Let G be a cyclic group generated by a and let II be a subgroup of G. If II = {e}, then H = (e) is cyclic. If H integer in Z+ such that a" (e), then a" H for some n € Z+. Let m be the smallest H. We claim that c = a" generates H; that is, We must show that every b e H is a power of c. Since be H and H ≤ G, we have b = a" for some n. Find q and r such that H = (a") = (c). n = mq + r for in accord with the division algorithm. Then SO 0≤r