Euler characteristics of centralizer subcategories

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.