Large orbits of $p$-groups on conjugacy classes

Abstract

Let $P$ be a $p$-group that acts faithfully and coprimely on a finite group $H$. A classical result of D.\ S.\ Passman, in the form proved by I.\ M.\ Isaacs, asserts that there exists an element $x\in H$ such that $|C_P(x)|\le (|P|/p)^{1/p}$. In this paper, we study the corresponding question for the action of $P$ on the conjugacy classes of $H$. We show that for every odd prime $p$ the analogue of Isaacs' bound fails for classes. On the other hand, we prove that if $H$ is solvable, then there always exists a conjugacy class $c$ of $H$ whose stabilizer in $P$ has order at most $|P|^{1/2}$, and that this bound is best possible for every prime $p$. As a consequence, there always exists an irreducible character of $H$ whose inertia group in $P$ has order at most $|P|^{1/2}$. The proof proceeds via a result on linear actions of $p$-nilpotent groups, and in the quasiprimitive case it relies on a classification of Holt and Yang and on computer calculations.