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Overview

This paper explores the deep connection between the Rubik’s Cube and abstract algebra, specifically Group Theory. It introduces group operations using cube moves and illustrates the structure of permutation groups.

We study the Rubik’s Cube group $G$ generated by the basic face turns $\mathcal{R}=\{F,R,L,U,D,B\}$. Each legal configuration corresponds to an element of $G$, with composition as the group operation. The eight corners and twelve edges carry orientation data. A convenient viewpoint embeds \(G\) into permutation groups on oriented pieces: \[ G \subseteq \mathrm{Sym}(\Omega). \]

• Corner orientations live in \(\mathbb{Z}_3^8\) with total twist \( \equiv 0 \pmod{3}\).
• Edge orientations live in \(\mathbb{Z}_2^{12}\) with total flip \( \equiv 0 \pmod{2}\).

Homomorphisms capture this: \( \phi_1: G \to S_{8}\ltimes \mathbb{Z}_3^8, \) \( \phi_2: G \to S_{12}\ltimes \mathbb{Z}_2^{12} \)

Commuting Generators

Generators with disjoint support commute, e.g. \(UD=DU\). More generally, if \(\mathrm{supp}(X)\cap \mathrm{supp}(Y)=\varnothing\) then \(XY=YX\).