Document Zbl 1294.20033

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L.</a>; <a href="/authors/parker.chris-w" title="Author Profile">Parker, C. W.</a></div> <h2 class="title"> <strong>The diameter of the commuting graph of a finite group with trivial centre.</strong> <i>(English)</i> <a class="label nowrap" href="/1294.20033">Zbl 1294.20033</a> </h2> <div class="source"> <a href="/serials/473" title="Journal Profile">J. Algebra</a> <a href="/?q=in%3A330494" title="Articles in this Issue">393, 41-59 (2013)</a>. </div> <div class="abstract">The commuting graph $\Gamma(G)$ of a group $G$ is the graph which has vertices the non-central elements of $G$ and two distinct vertices of $\Gamma(G)$ are adjacent if and only if they commute in $G$.<br class="zbmathjax-paragraph"> <span class="zbmathjax-textit">A. Iranmanesh</span> and <span class="zbmathjax-textit">A. Jafarzadeh</span> [J. Algebra Appl. 7, No. 1, 129-146 (2008; <a href="/1151.20013">Zbl 1151.20013</a>)] demonstrated that the commuting graphs of the finite symmetric and alternating groups are either disconnected or have diameter at most 5 and conjectured that there is an absolute upper bound for the diameter of a connected commuting graph of a non-Abelian finite group. <span class="zbmathjax-textit">M. Giudichi</span> and <span class="zbmathjax-textit">C. Parker</span> [J. Comb. Theory, Ser. A 120, No. 7, 1600-1603 (2013)] have shown that this conjecture is incorrect by constructing an infinite family of special 2-groups with commuting graphs of increasing diameter. In this paper, the authors prove that the connected components of the commuting graph of a finite group with trivial centre have diameter at most 10.<br class="zbmathjax-paragraph"> Various special kinds of finite groups with bounded diameter commuting graph were investigated by <span class="zbmathjax-textit">Y. Segev</span> and <span class="zbmathjax-textit">G. M. Seitz</span> [Pac. J. Math. 202, No. 1, 125-225 (2002; <a href="/1058.20015">Zbl 1058.20015</a>)], Iranmanesh and Jafarzadeh [loc. cit.], <span class="zbmathjax-textit">T. J. Woodkock</span> [Commuting graphs of finite groups. PhD thesis, Univ. Virginia (2010)], <span class="zbmathjax-textit">M. Giudici</span> and <span class="zbmathjax-textit">A. Pope</span> [Australas. J. Comb. 48, 221-230 (2010; <a href="/1232.05114">Zbl 1232.05114</a>); J. Group Theory 17, No. 1, 131-149 (2014)], the second author [Bull. Lond. Math. Soc. 45, No. 4, 839-848 (2013; <a href="/1278.20017">Zbl 1278.20017</a>)].<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A118">Anatoli Kondrat’ev (Ekaterinburg)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6267897">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A6267897">Cited in <strong>45</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%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%3A20D05" title="MSC2020">20D05</a> </td> <td class="space"> Finite simple groups and their classification </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C40" title="MSC2020">05C40</a> </td> <td class="space"> Connectivity </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afinite+groups">finite groups</a>; <a href="/?q=ut%3Asimple+groups">simple groups</a>; <a href="/?q=ut%3Acommuting+graphs">commuting graphs</a>; <a href="/?q=ut%3Adiameters">diameters</a>; <a href="/?q=ut%3Aconnected+graphs">connected graphs</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1151.20013">Zbl 1151.20013</a>; <a href="/1058.20015">Zbl 1058.20015</a>; <a href="/1232.05114">Zbl 1232.05114</a>; <a href="/1278.20017">Zbl 1278.20017</a></div> <div class="software"> <h3>Software:</h3><a href="/software/540">Magma</a>; <a href="/software/13584">ATLAS Group Representations</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 1294.20033" data-ciurl="/ci/06267897" data-biburl="/bibtex/06267897.bib" data-amsurl="/amsrefs/06267897.bib" data-xmlurl="/xml/06267897.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06267897.pdf" title="Zbl 1294.20033 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/10.1016/j.jalgebra.2013.06.031" aria-label="DOI for “The diameter of the commuting graph of a finite group with trivial centre.”" title="10.1016/j.jalgebra.2013.06.031">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1301.2341"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Akbari, S.; Mohammadian, A.; Radjavi, H.; Raja, P., On the diameters of commuting graphs, Linear Algebra Appl., 418, 161-176 (2006) · <a href="/1107.05056" class="nowrap">Zbl 1107.05056</a></td> </tr><tr> <td>[2]</td> <td class="space">Aschbacher, M., Finite Group Theory, Cambridge Stud. 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J., Commuting graphs of finite groups (2010), University of Virginia, PhD thesis</td> </tr><tr> <td>[29]</td> <td class="space">Zavarnitsin, A. V., On the recognition of finite groups by the prime graph, Algebra Logika. Algebra Logika, Algebra Logic, 45, 4, 220-231 (2006), 502 (in Russian); translation in: · <a href="/1115.20011" class="nowrap">Zbl 1115.20011</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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