Document Zbl 1530.05079

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Algebr. Comb.</a> <a href="/?q=in%3A506602" title="Articles in this Issue">58, No. 4, 1095-1124 (2023)</a>. </div> <div class="abstract">Summary: We prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classified by <span class="zbmathjax-textit">P. Manna</span> et al. [Electron. J. Comb. 28, No. 3, Research Paper P3.4, 14 p. (2021; <a href="/1467.05115">Zbl 1467.05115</a>)]. Here we classify the finite simple groups with chordal power graph, relative to typical number theoretic conditions. We do so by devising several sufficient conditions for the existence and non-existence of long cycles in power graphs of finite groups. We examine other natural group classes, including special linear, symmetric, generalized dihedral and quaternion groups, and we characterize direct products with chordal power graph. The classification problem is thereby reduced to directly indecomposable groups, and we further obtain a list of possible socles. Lastly, we give a general bound on the length of an induced path in chordal power graphs, providing another potential road to advance the classification beyond simple groups.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C25" title="MSC2020">05C25</a> </td> <td class="space"> Graphs and abstract algebra (groups, rings, fields, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C38" title="MSC2020">05C38</a> </td> <td class="space"> Paths and cycles </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C35" title="MSC2020">05C35</a> </td> <td class="space"> Extremal problems in graph theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20D60" title="MSC2020">20D60</a> </td> <td class="space"> Arithmetic and combinatorial problems involving abstract finite groups </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20D05" title="MSC2020">20D05</a> </td> <td class="space"> Finite simple groups and their classification </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Apower+graph">power graph</a>; <a href="/?q=ut%3Achordal+graph">chordal graph</a>; <a href="/?q=ut%3Asimple+groups">simple groups</a>; <a href="/?q=ut%3Adirect+products">direct products</a>; <a href="/?q=ut%3Amaximal+cycles+in+power+graphs">maximal cycles in power graphs</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1467.05115">Zbl 1467.05115</a></div> <div class="software"> <h3>Software:</h3><a href="/software/13584">ATLAS Group Representations</a>; <a href="/software/825">SageMath</a>; <a href="/software/320">GAP</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1530.05079" data-ciurl="/ci/07784772" data-biburl="/bibtex/07784772.bib" data-amsurl="/amsrefs/07784772.bib" data-xmlurl="/xml/07784772.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07784772.pdf" title="Zbl 1530.05079 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s10801-023-01262-2" aria-label="DOI for “On groups with chordal power graph, including a classification in the case of finite simple groups”" title="10.1007/s10801-023-01262-2">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2209.00317"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Aalipour, G., Akbari, S., Cameron, P.J., Nikandish, R., Shaveisi, F.: On the structure of the power graph and the enhanced power graph of a group. 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