Let $p$ be a prime and let $X$ be a finite set whose cardinality is divisible by $p$. Assume a group $G$ of order $p^n$ for $n>1$ acts on $X$. Show that if $G$ fixes an element in $X$ then it fixes more than one element.
No idea with this problem. Maybe there is some useful key facts about such kind of group. May I please ask to pointing them out or prove it directly? Many thanks!