On $\operatorname{Alt}(n)$-modules with an additive dimension when $n\le6$

TL;DR

This paper determines the minimal dimension of faithful Alt(n)-modules in general and classifies specific low-dimensional cases in certain characteristics, advancing the understanding of modules with additive dimensions.

Contribution

It completes the classification of minimal faithful Alt(n)-modules and details specific classifications for certain exceptional cases.

Findings

Classified 2D Alt(5)-modules in characteristic 2

Classified 3D Alt(5)-modules in characteristic 5

Classified 3D Alt(6)-modules in characteristic 3

Abstract

Working in the general context of "modules with an additive dimension," we complete the determination of the minimal dimension of a faithful Alt(n)-module and classify those modules in three of the exceptional cases: 2-dimensional Alt(5)-modules in characteristic 2, 3-dimensional Alt(5)-modules in characteristic 5, and 3-dimensional Alt(6)-modules in characteristic 3. We also highlight the remaining work needed to complete the classification of the faithful Alt(n)-modules of minimal dimension for all n; these open problems seem well suited as projects for advanced undergraduate or master's students.