# Thread: Relatively Prime Proof

1. ## Relatively Prime Proof

Assume that a is an element of order n in a group G. Prove that m and n are relatively
prime if and only if a^m has order n.

Thanks for your help...

2. You don't need to prove anything, this follows from the definitions: If $a$ is an element of order $n$ in a group and $m$ is a positive integer, then $=$ and $|a^m|=\frac{n}{gcd(n,m)}$. So if n and m are relatively prime then $|a^m|= \frac{n}{gcd(m,n)}=\frac{n}{1}=n$.

To see why that definition holds, let d=gcd(n,m), clearly $(a^d)^{n/d}=a^n=e$, so that | $a^d| \leq n/d$. Also if $i$ is a positive integer less than n/d. then $(a^d)^i \ne e$ by definition of |a|. And since d=gcd(n,m) you get $|a^m|=||=||=|a^{gcd(n,m)}| = \frac{n}{gcd(n,m)}$.