Euler characteristics of centralizer subcategories

TL;DR

This paper studies the equivariant Euler characteristics of centralizer subcategories in finite groups, establishing divisibility properties and verifying conjectures through explicit calculations for Mathieu groups.

Contribution

It introduces a global relation linking local Euler characteristics to p-elements centralizers and confirms the equivariant Brown's theorem and Knorr-Robinson conjecture in specific cases.

Findings

The reduced Euler characteristic of the equivariant Brown poset is divisible by the p-part of the centralizer order.

A concrete numerical verification of the Knorr-Robinson conjecture for the Mathieu group.

Establishment of a global relation between local Euler characteristics and centralizer p-elements.

Abstract

Let p be a prime number, G a finite group, and A a finite group acting on G. The Brown poset of nonidentity p-subgroups of G is then an A-poset. We investigate the equivariant subposet and the equivariant Euler characteristics and establish a global relation between locally defined Euler characteristics and the number of p-elements of G centralized by A. It is a consequence of this relation that the equivariant version of Brown's theorem holds: The reduced Euler characteristic of the A-equivariant Brown poset is divisible by the p-part of the order of the centralizer of A. The second equivariant Euler characteristic for the conjugation of G on the Brown poset for G is especially intriguing because of its relation to the Knorr-Robinson conjecture and we carry out a concrete numerical verification of the conjecture in case of the smallest simple Mathieu group.