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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.05648 |
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| _version_ | 1866918171670216704 |
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| author | Cunningham, Gabe Minevich, Igor |
| author_facet | Cunningham, Gabe Minevich, Igor |
| contents | The Lights Out Puzzle, played on a graph $Γ$, has been studied using linear algebra over $\mathbb{F}_2$ and more generally over $\mathbb{Z}/k\mathbb{Z}$. We generalize the setting by allowing the states of vertices to be the elements of a group $G$, where a \textit{click} in vertex $v$ multiplies the state of $v$ and its neighbors by an element $g \in G$ on the right.
Starting with the identity element $e \in G$ for all vertices, the totality of all achievable state configurations forms a group $G^Γ$. 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 $\mathbb{Z}$. We discuss a chain of five different subgroups consisting of commutators and introduce techniques for showing that families of graphs are RA using each. In particular, using Heisenberg groups, we establish that a graph is RA precisely when a certain lattice spans $\mathbb{Z}^{|Γ|}$. While most graphs appear to be RA, we show the odd-dimensional cube graphs $Q_{2n+1}$ and folded cube graphs $\square_d$, for $d$ odd or 2, are not. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05648 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graph Powers of Groups Cunningham, Gabe Minevich, Igor Group Theory Combinatorics 20-02 (Primary) 20-04, 05C30 (Secondary) The Lights Out Puzzle, played on a graph $Γ$, has been studied using linear algebra over $\mathbb{F}_2$ and more generally over $\mathbb{Z}/k\mathbb{Z}$. We generalize the setting by allowing the states of vertices to be the elements of a group $G$, where a \textit{click} in vertex $v$ multiplies the state of $v$ and its neighbors by an element $g \in G$ on the right. Starting with the identity element $e \in G$ for all vertices, the totality of all achievable state configurations forms a group $G^Γ$. 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 $\mathbb{Z}$. We discuss a chain of five different subgroups consisting of commutators and introduce techniques for showing that families of graphs are RA using each. In particular, using Heisenberg groups, we establish that a graph is RA precisely when a certain lattice spans $\mathbb{Z}^{|Γ|}$. While most graphs appear to be RA, we show the odd-dimensional cube graphs $Q_{2n+1}$ and folded cube graphs $\square_d$, for $d$ odd or 2, are not. |
| title | Graph Powers of Groups |
| topic | Group Theory Combinatorics 20-02 (Primary) 20-04, 05C30 (Secondary) |
| url | https://arxiv.org/abs/2502.05648 |