Abstract 3D-rotation groups and recognition of icosahedral modules

Abstract

We introduce an abstract notion of a 3D-rotation module for a group $G$ that does not require the module to carry a vector space structure, a priori nor a posteriori. We prove that, under an expected irreducibility-like assumption, the only finite $G$ with such a module are those already known from the classical setting: $\operatorname{Alt}(4)$, $\operatorname{Sym}(4)$, and $\operatorname{Alt}(5)$. Our main result then studies the module structure when $G = \operatorname{Alt}(5)$ and shows that, under certain natural restrictions, it is fully determined and generalizes that of the classical icosahedral module. We include an application to the recently introduced setting of modules with an additive dimension, a general setting allowing for simultaneous treatment of classical representation theory of finite groups as well as representations within various well-behaved model-theoretic settings such as the $o$-minimal and finite Morley rank ones. Leveraging our recognition result for icosahedral modules, we classify the faithful $\operatorname{Alt}(5)$-modules with additive dimension that are dim-connected of dimension $3$ and without $2$-torsion.