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

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 $\mathbb{R}$.