A 2-block splitting in alternating groups

Abstract

In 1956, Brauer showed that there is a partitioning of the $p$-regular conjugacy classes of a group according to the $p$-blocks of its irreducible characters with close connections to the block theoretical invariants. In a previous paper, the first explicit block splitting of regular classes for a family of groups was given for the 2-regular classes of the symmetric groups. Based on this work, the corresponding splitting problem is investigated here for the 2-regular classes of the alternating groups. As an application, an easy combinatorial formula for the elementary divisors of the Cartan matrix of the alternating groups at $p=2$ is deduced.