Modular representations of metabelian groups

Let $M$ be a module over a commutative ring with unity $R$, and let $ϕ:M \to R$ be an $R$-linear map. Then the product $mn = m + (1+ϕ(m))n$ defines a monoid $Mon(ϕ)$ on $M$. We show that if $G$ is a group, then $G$ arises as the group of invertible elements of some such monoid $Mon(ϕ)$ if and only if $G$ is metabelian.

Authors

Publication Details

Journal
arXiv (Cornell University)
Published
2026-10-26
DOI
https://doi.org/10.48550/arxiv.2610.35603
Primary Topic
Finite Group Theory Research
Type
preprint
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preprint

Modular representations of metabelian groups

Lorenzo Traldi
arXiv (Cornell University)
Finite Group Theory Research
preprint

Modular representations of metabelian groups

Lorenzo Traldi
preprint en

Abstract

Let $M$ be a module over a commutative ring with unity $R$, and let $ϕ:M \to R$ be an $R$-linear map. Then the product $mn = m + (1+ϕ(m))n$ defines a monoid $Mon(ϕ)$ on $M$. We show that if $G$ is a group, then $G$ arises as the group of invertible elements of some such monoid $Mon(ϕ)$ if and only if $G$ is metabelian.

arXiv (Cornell University)
Finite Group Theory Research
DOIOpen Access PDFOpenAlex
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Modular representations of metabelian groups — Lorenzo Traldi · arXiv (Cornell University) (2026) | TGRS Research Map | TGRS