Large orbits in actions of nilpotent groups

I. M. Isaacs
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, 
Madison, Wisconsin 53706 

Proc. Amer. Math. Soc. 127 (1999), 45-50. 

Abstract. If a nontrivial nilpotent group $N$ acts faithfully and coprimely 
on a group $H$, it is shown that some element of $H$ has a small centralizer 
in $N$ and hence lies in a large orbit. Specifically, there exists $x \in H$ 
such that $|C_N(x)| \le (|N|/p)^{1/p}$, where $p$ is the smallest prime 
divisor of $|N|$.