# Math Help - Conjugates in a finite group

1. ## Conjugates in a finite group

Show that in a finite group of order n, an element of order k has at most n/k conjugates.

Proof so far.

Let G be a group with order n, and suppose that g is an element with the order of k. Then I have k|n by the property of order. But I'm not too familiar in dealing with conjugates, how should I get started here? Thanks.

Let $G$ act on itself by conjugative. Then the conjugates to $a\in G$ is the orbit $Ga$. But $|Ga| = |G:G_a|$. And $|G_a| = |C(a)|$ - the centralizer. But $\{ a,a^2,...,a^k\} \subseteq C(a)$. Therefore, $|C(a)| \geq k$. This means $[G:G_a] \leq n/k$. Therefore, $|Ga| \leq n/k$.