Conjugacy class numbers and $\pi$-subgroups

Gunter Malle, Gabriel Navarro and Geoffrey R. Robinson

Vol. 311 (2021), No. 1, 135–164

DOI: 10.2140/pjm.2021.311.135

Abstract

The number $l\left(G\right)$ of $p$-regular classes of a finite group $G$ is a key invariant in modular representation theory. Several outstanding conjectures propose that this number can be calculated or bounded in terms of certain invariants of some subgroups of $G$. Our main question here is if $l\left(G\right)$ can be bounded by the number of conjugacy classes of some subgroup of $G$ of order not divisible by $p$. This would have consequences for the Malle–Robinson $l\left(B\right)$-conjecture. Furthermore, we investigate a $\pi$-version of this, for sets of primes $\pi$.

As part of our investigations, we study finite groups that have more conjugacy classes than any of their proper subgroups. These groups naturally appear in questions on bounding from above the number of conjugacy classes of a group, and were considered by G. R. Robinson and J. G. Thompson in the context of the $k\left(GV\right)$-problem. We classify the almost Abelian groups $G$ with $F^{\ast }\left(G\right)$ quasisimple. Our results should be of use in several related questions.

Keywords

number of conjugacy classes, $\pi$-subgroups, almost Abelian finite groups, $l(B)$-conjecture

Mathematical Subject Classification

Primary: 20C20, 20D20, 20D60

Milestones

Received: 26 May 2020

Revised: 8 January 2021

Accepted: 12 January 2021

Published: 17 March 2021