Give at most a three-sentence synopsis of the proof of Theorem 6.6. 6.6 Theorem A subgroup of
Question:
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.
Transcribed Image Text:
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
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
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