# cyclic subgroup

Let G = <a> be cyclic group of order n.

a) Prove that the cyclic subgroup generated by a^m is the same as the cyclic subgroup generated by a^d, where d = (m,n)

b) Prove that a^m is a generator of G if and only if (m,n) = 1

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#### Solution Preview

Proof:

(a) Since d = (m,n), then we can find two integer q and r such that mq + nr = d. Since |G| = n, then a^n = e.

Then we have a^d = a^(mq + nr) = a^(mq) = (a^m)^q is in <a^m>. So ...

#### Solution Summary

Cyclic subgroups are contextualized.

$2.19