Solomon's induction in quasi-elementary groups

TL;DR

This paper investigates the conditions under which multiples of the trivial representation in finite groups can be expressed as inductions from proper subgroups, focusing on quasi-elementary groups and extending Solomon's induction theorem.

Contribution

It extends Solomon's induction theorem by characterizing when multiples of p are induced from proper subgroups in p-quasi-elementary groups, establishing the optimality of these conditions.

Findings

All multiples are induced if G is not quasi-elementary.

All multiples of p are induced if G is p-quasi-elementary and not cyclic.

The results are proven to be optimal.

Abstract

Given a finite group G, we address the following question: which multiples of the trivial representation are linear combinations of inductions of trivial representations from proper subgroups of G? By Solomon's induction theorem, all multiples are if G is not quasi-elementary. We complement this by showing that all multiples of p are if G is p-quasi-elementary and not cyclic, and that this is best possible.