Today we learned about subgroups of cyclic groups G = <a>. During the discussion we reached this point:(adsbygoogle = window.adsbygoogle || []).push({});

|<a^k>| = minimum L, L > 0, such that a^(kL) = 1.

|G| = n.

Then a^kL = a^bn, thus kL = bn, and thusL = n/gcd(k, n).

However, I don't understand the bolded. My number theory is terrible, and I don't really see where the gcd(k, n) comes from.

I understand that if gcd(k, n) = 1 that <a^k> = <a>, but the connection to L = n/gcd(k,n) just isn't apparent to me. Can someone shine some light on this for me?

Thanks, I appreciate it!

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# The order of cyclic subgroups

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