{
  "abstract": "The Lights Out Puzzle, played on a graph $\\Gamma$, 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 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 $G^{\\Gamma}$. 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 $\\mathbb{Z}$. 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 $\\mathbb{Z}^{|\\Gamma|}$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "Gabe Cunningham"
    },
    {
      "name": "Igor Minevich"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2a0qw26ds573r.bib",
  "cited_by": 0,
  "doi": "10.37236/14315",
  "fulltext_url": null,
  "id": "2a0qw26ds573r",
  "issns": [
    "1077-8926"
  ],
  "landing_url": "https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p16",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W4407385203",
  "pdf_url": "https://www.combinatorics.org/ojs/index.php/eljc/article/download/v33i4p16/pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": "The Electronic Journal of Combinatorics",
  "ris": "https://research.draconitic.com/paper/2a0qw26ds573r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:24Z",
      "native_id": "10.37236/14315",
      "source": "crossref",
      "url": "https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p16"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2607|Finite Group Theory Research",
      "subfield": "combinatorics"
    }
  ],
  "title": "Graph Powers of Groups",
  "title_text": "Graph Powers of Groups",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:24Z",
  "url": "https://research.draconitic.com/paper/2a0qw26ds573r",
  "venue": "The Electronic Journal of Combinatorics",
  "versions": []
}