Graph Products of Groups

Abstract: Imagine three Rubik's Cubes arranged in a row, linked such that any move on the left or right cube also applies to the middle one, and moves on the middle cube affect all three simultaneously. If each cube is individually solvable, can all three be solved together in this configuration? This question motivates a natural generalization of the Lights Out Puzzle, exploring the synergy between graph theory and group theory. The classic Lights Out Puzzle is played on a $5 \times 5$ grid where each square is either "on" or "off." Clicking a square toggles its state and that of its neighbors, and the objective is to turn all squares "off." A related question, solvable using linear algebra over $\mathbb{F}2$, the finite field of 2 elements, asks which states are possible to achieve starting from the all-on state. Here, we vastly generalize the setting in two ways: (1) the states are allowed to be the elements of a group $G$, where "clicking" is done by selecting an element $g \in G$ and multiplying all affected squares on the right by $g$, and (2) we not only allow square grids, but any graph $\Gamma$, where clicking a vertex $v$ affects $v$ and all adjacent vertices. Starting with the identity element $e \in G$ for all vertices, the totality of all achievable state configurations forms a group $G^{\Gamma}$. This group generalizes parallel products of group actions and provides a rich structure for analysis. For many graphs, which we term "RA" (reducible to abelian), the problem reduces - regardless of $G$ - to a linear algebra question over $\mathbb{Z}$ (the integers). We discuss a chain of five different subgroups consisting of commutators; the largest two are equal precisely when the graph is RA. Interestingly, while most graphs appear to be RA, we show the odd-dimensional cube graphs $Q{2n+1}$ are not.