Suppose $G$ is finite group of order $p^dn$ where $d$ and $n$ are positive integers and $p$ is a prime that does not divide $n$. Show that $G$ contains an element of order $p$ such that the cardinality of its conjugacy class divides $n$
Cardinality of conjugacy class of an element divides the order of group. So it is sufficient to prove that for some element $g$ of order $p$, $p^d\mid C(g)$ where $C(g)$ is the centralizer.
Please give a hint. Please do not give solution. Thanks!
Edit: I realized that the above attempt does not make use of (i thought it does make use of) the fact that $p$ does not divide $n$.