Computing the projective indecomposable modules of large finite groups

Abstract

Let G be a finite group and [special characters omitted] be a finite field. A projective indecomposable [special characters omitted]G-module is an indecomposable direct summand of the group algebra [special characters omitted]G. Computing the projective indecomposable modules of large finite groups has been always a challenging problem due to the large sizes of the representations of these groups. This dissertation describes a new algorithm for constructing the projective indecomposable modules of large finite groups. This algorithm uses the condensation techniques as described in [12]. The power of the algorithm will be illustrated by the examples of the socle series of all projective indecomposable modules of the sporadic simple Mathieu group M₂₄ and the simple alternating group A₁₂ in characteristic 2.