Abstract 3D-rotation groups and recognition of icosahedral modules

TL;DR

This work extends Klein’s classical classification of finite subgroups of SO(3) to a broader, non-vector-space context by introducing 3D-rotation modules for groups acting on groups. Under a natural irreducibility-like condition, the finite groups admitting nonaxial 3D-rotation modules are exactly Alt(4), Sym(4), and Alt(5), with their axis centralizers cyclic. Focusing on G = Alt(5), the authors prove an explicit recognition theorem: under mild technical hypotheses, the nonaxial Alt(5)-module is icosahedral, realized as Icosa(R) with R = Z[phi], phi^2 = phi + 1, and further extended to Icosa(L) for 2-torsion-free L. They connect this picture to modules with an additive dimension, obtaining corollaries that classify minimal Alt(5)-modules in the additive-dimension framework and providing examples like Icosa(C_p∞) when p ≡ ±1 (mod 5). The results bridge classical finite group representation theory with model-theoretic contexts such as finite Morley rank and o-minimal settings, offering a robust, constructive route to recognizing icosahedral structure in a broad algebraic environment.

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.

Figures (1)

• Figure 1: Conjugacy class of $\sigma_1$: $\mathcal{C}_{5,1}$

Theorems & Definitions (66)

• Definition
• Remark
• Theorem 1: Generalizing Klein
• Definition : Icosahedral Modules
• Remark
• Example 1.1
• Theorem 2
• Corollary 2.1
• Corollary 2.2
• Lemma 2.1: cf. BrE23