Testing Modules of Groups of Even Order for Simplicity,

In this paper we exhibit an intimate relationship between the simplicity of anFG-moduleVof a finite groupGgenerated by a noncentral involutiont ≠ 1 and some other elementuinGand the socles of the eigenspacesEλfor λ in { − 1, 1} of the involutiontconsidered asFC-modules, whereC = CG(t) is the centralizer oftinG. In fact the two main results yield new simplicity criteria forFG-modulesVover fieldsFwith odd or even characteristicp, respectively. Forp ≠ 2 the result follows from a general module theoretic simplicity test proved in the first section of the paper. It builds on ideas of the Meat-axe algorithm of previous work. We show the practicability of our new tests by examples.