On $p$-parts of conjugacy class sizes and the index of the $p$-core

TL;DR

The paper addresses bounding the $p$-part of the index $|G:\mathbf{O}_{p}(G)|$ under a conjugacy-class size constraint for $p$-regular elements, specifically when $p^{2}$ does not divide the size of any such conjugacy class. It extends Itô–Michler-type questions by developing the $\mathbf{SCpR}(p^{n})$ framework and applying it to the dual setting of conjugacy classes, aided by block theory and character-height considerations. The main result shows $|G:\mathbf{O}_{p}(G)|_p\le p^{2}$ under the hypothesis, and this bound is sharp, as illustrated by a standard semilinear-group construction with $\mathbf{O}_{p}(G)=1$ and $|G|_p=p^{2}$. The proofs combine minimal-counterexample arguments, inheritance of $\mathbf{SCpR}(p^{2})$ under normal subgroups and quotients, radical reductions, and defect-group/height techniques from Brauer theory to reach the conclusion.

Abstract

For a prime $p$ and an arbitrary finite group $G$, we show that if $p^{2}$ does not divide the size of each conjugacy class of \emph{$p$-regular} element (element of order not divisible by $p$) in $G$, then the largest power of $p$ dividing the index $|G:\mathbf{O}_{p}(G)|$ is at most $p^{2}$.

Theorems & Definitions (6)

• Theorem 1
• proof
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['thmA']}