A note on the projective representations of $p$-solvable and $\pi$-separable group

TL;DR

This paper extends the Itô--Michler's Theorem to projective representations of p-solvable and π-separable groups, linking their irreducible modules to those of Sylow p- and Hall π-subgroups.

Contribution

It completes the theorem for these classes of groups and relates their projective irreducible modules to subgroups, advancing understanding of their representation theory.

Findings

Extended Itô--Michler's Theorem to p-solvable and π-separable groups

Established correspondence between projective irreducible modules and subgroup modules

Provided new insights into the structure of projective representations

Abstract

Following Gluck and Wolf we complete the It\^o--Michler's Theorem for the projective representations of a $p$-solvable or $\pi$-separable group, and then we relate the projective irreducible modules of such a group with those of its Sylow $p$-subroups and Hall $\pi$-subgroups.