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© BrainMass Inc. brainmass.com October 10, 2019, 2:17 am ad1c9bdddf
(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 ...
Cyclic subgroups are contextualized.