The Rubik's Cube and Minimal Representations of Split Group Extensions

TL;DR

This paper analyzes the algebraic structure of groups associated with the Rubik's Cube, providing new insights into their embeddings and minimal faithful representations over complex and real fields.

Contribution

It introduces a split homomorphism between the groups for 2x2 and 3x3 cubes and determines their minimal faithful representation dimensions over d6 and a7.

Findings

G_2 embeds into G_3 as a subgroup

Minimal faithful dimension of G_2 is 8 over d6 and 16 over a7

Minimal faithful dimension of G_3 is 20 over d6 and 28 over a7

Abstract

In this paper, we examine the groups $G_2$ and $G_3$ associated to the $2 \times 2$ and $3 \times 3$ Rubik's cubes. We express $G_2$ and $G_3$ in terms of familiar groups and exhibit a split homomorphism $\psi: G_3 \longrightarrow G_2$ to prove that $G_2$ embeds inside $G_3$ as a subgroup. In addition, we prove several results bounding the dimensions of minimal faithful representations of finite abelian groups split by some complementary subgroup. We then employ these results to determine the minimal faithful dimensions of $G_2$ and $G_3$ over both $\mathbb{C}$ and $\mathbb{R}$. We find that $G_2$ has minimal dimension 8 over $\mathbb{C}$ and 16 over $\mathbb{R}$, and that $G_3$ has minimal dimension 20 over $\mathbb{C}$ and 28 over $\mathbb{R}$.