Finite Groups with Submultiplicative Spectra

TL;DR

This paper investigates finite groups with property $ ilde{s}$, where all subrepresentations have submultiplicative spectra, revealing their nilpotent nature and characterizing specific classes like abelian, regular, and V-regular groups.

Contribution

It characterizes finite groups with property $ ilde{s}$, especially $p$-groups, and establishes conditions under which they are abelian, regular, or V-regular, advancing understanding of their spectral properties.

Findings

2-groups with property $ ilde{s}$ are exactly the abelian groups.

All $p$-abelian groups have property $ ilde{s}$, including groups of exponent $p$.

Certain metabelian $p$-groups are V-regular if and only if they have property $ ilde{s}$.

Abstract

We study abstract finite groups with the property, called property $\hat{s}$, that all of their subrepresentations have submultiplicative spectra. Such groups are necessarily nilpotent and we focus on $p$-groups. $p$-groups with property $\hat{s}$ are regular. Hence, a 2-group has property $\hat{s}$ if and only if it is commutative. For an odd prime $p$, all $p$-abelian groups have property $\hat{s}$, in particular all groups of exponent $p$ have it. We show that a 3-group or a metabelian $p$-group ($p \ge 5$) has property $\hat{s}$ if and only if it is V-regular.