On groups with Cayley graph isomorphic to a cube

TL;DR

This paper characterizes groups whose Cayley graphs are cubes, showing they decompose into products of involutions and have reducible geometric representations, enriching understanding of cube-like symmetries in group theory.

Contribution

It introduces the concept of cube groups, provides a combinatorial decomposition into involution products, and proves the reducibility of their geometric representations.

Findings

Cube groups are generated by involutions with Cayley graphs isomorphic to a cube.

They decompose into products of 2-element subgroups.

The geometric representation of cube groups is always reducible.

Abstract

We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is simply-transitive on the vertices and the edge stabilizers are all nontrivial. The action on the cube extends to an orthogonal linear action, which we call the geometric representation. We prove a combinatorial decomposition for cube groups into products of 2-element subgroup, and show that the geometric representation is always reducible.