Document Zbl 1112.20006

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Am. Math. Soc.</a> <a href="/?q=in%3A188211" title="Articles in this Issue">135, No. 3, 671-676 (2007)</a>. </div> <div class="abstract">Let $G$ be a finite group and let $\text{cd}(G)$ denote the different integers which occur as the degrees of the irreducible complex characters of $G$. Let $\Gamma(G)$ be the graph whose vertex set is $\text{cd}(G)-\{1\}$. An edge joins two vertices represented by positive integers $a$ and $b$ if $\gcd(a,b)>1$.<br class="zbmathjax-paragraph"> The authors prove that if $\Gamma(G)$ is a complete graph, then $G$ is solvable.<br class="zbmathjax-paragraph"> The proof, as might be anticipated, uses the classification of finite simple groups, and divides itself into three parts. For groups of Lie type, properties of the Steinberg character are used. For alternating groups, two irreducible characters whose degrees are small and relatively prime, and which extend to the symmetric group, are used. For sporadic groups, pairs of irreducible characters of relatively prime degrees, which also extend to the relevant automorphism group, are employed. These are found using the ATLAS.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A8404">Roderick Gow (Dublin)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5120273">Cited in <strong>2</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5120273">Cited in <strong>118</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A20C15" title="MSC2020">20C15</a> </td> <td class="space"> Ordinary representations and characters </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%3A20D60" title="MSC2020">20D60</a> </td> <td class="space"> Arithmetic and combinatorial problems involving abstract finite groups </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acharacter+degrees">character degrees</a>; <a href="/?q=ut%3Acomplete+graphs">complete graphs</a>; <a href="/?q=ut%3Asimple+groups">simple groups</a>; <a href="/?q=ut%3Airreducible+complex+characters">irreducible complex characters</a>; <a href="/?q=ut%3Asolvability+of+groups">solvability of groups</a></div> <div class="software"> <h3>Software:</h3><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 1112.20006" data-ciurl="/ci/05120273" data-biburl="/bibtex/05120273.bib" data-amsurl="/amsrefs/05120273.bib" data-xmlurl="/xml/05120273.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05120273.pdf" title="Zbl 1112.20006 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/10.1090/S0002-9939-06-08651-5" aria-label="DOI for “Character degree graphs that are complete graphs.”" title="10.1090/S0002-9939-06-08651-5">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. · <a href="/0568.20001" class="nowrap">Zbl 0568.20001</a></td> </tr><tr> <td>[2]</td> <td class="space">Yakov Berkovich, Finite groups with small sums of degrees of some non-linear irreducible characters, J. Algebra 171 (1995), no. 2, 426 – 443. · <a href="/0829.20011" class="nowrap">Zbl 0829.20011</a> · <a href="https://doi.org/10.1006/jabr.1995.1020" class="nowrap">doi:10.1006/jabr.1995.1020</a></td> </tr><tr> <td>[3]</td> <td class="space">Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. · <a href="/0567.20023" class="nowrap">Zbl 0567.20023</a></td> </tr><tr> <td>[4]</td> <td class="space">Bertram Huppert, Character theory of finite groups, De Gruyter Expositions in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. · <a href="/0932.20007" class="nowrap">Zbl 0932.20007</a></td> </tr><tr> <td>[5]</td> <td class="space">I. Martin Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. · <a href="/0337.20005" class="nowrap">Zbl 0337.20005</a></td> </tr><tr> <td>[6]</td> <td class="space">I. M. Isaacs, Character correspondences in solvable groups, Adv. in Math. 43 (1982), no. 3, 284 – 306. · <a href="/0487.20004" class="nowrap">Zbl 0487.20004</a> · <a href="https://doi.org/10.1016/0001-8708(82)90037-8" class="nowrap">doi:10.1016/0001-8708(82)90037-8</a></td> </tr><tr> <td>[7]</td> <td class="space">Mark L. Lewis, Alexander Moretó, and Thomas R. Wolf, Non-divisibility among character degrees, J. Group Theory 8 (2005), no. 5, 561 – 588. · <a href="/1097.20011" class="nowrap">Zbl 1097.20011</a> · <a href="https://doi.org/10.1515/jgth.2005.8.5.561" class="nowrap">doi:10.1515/jgth.2005.8.5.561</a></td> </tr><tr> <td>[8]</td> <td class="space">M. L. LEWIS, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, to appear in Rocky Mountain J. Math. · <a href="/1166.20006" class="nowrap">Zbl 1166.20006</a></td> </tr><tr> <td>[9]</td> <td class="space">John K. McVey, Bounding graph diameters of nonsolvable groups, J. Algebra 282 (2004), no. 1, 260 – 277. · <a href="/1070.20009" class="nowrap">Zbl 1070.20009</a> · <a href="https://doi.org/10.1016/j.jalgebra.2004.08.019" class="nowrap">doi:10.1016/j.jalgebra.2004.08.019</a></td> </tr><tr> <td>[10]</td> <td class="space">Peter Schmid, Rational matrix groups of a special type, Linear Algebra Appl. 71 (1985), 289 – 293. · <a href="/0575.20005" class="nowrap">Zbl 0575.20005</a> · <a href="https://doi.org/10.1016/0024-3795(85)90255-1" class="nowrap">doi:10.1016/0024-3795(85)90255-1</a></td> </tr><tr> <td>[11]</td> <td class="space">Peter Schmid, Extending the Steinberg representation, J. Algebra 150 (1992), no. 1, 254 – 256. · <a href="/0794.20022" class="nowrap">Zbl 0794.20022</a> · <a href="https://doi.org/10.1016/S0021-8693(05)80060-2" class="nowrap">doi:10.1016/S0021-8693(05)80060-2</a></td> </tr><tr> <td>[12]</td> <td class="space">Alexandre Turull, Generic fixed point free action of arbitrary finite groups, Math. Z. 187 (1984), no. 4, 491 – 503. · <a href="/0574.20013" class="nowrap">Zbl 0574.20013</a> · <a href="https://doi.org/10.1007/BF01174184" class="nowrap">doi:10.1007/BF01174184</a></td> </tr><tr> <td>[13]</td> <td class="space">Donald L. White, Degree graphs of simple groups of exceptional Lie type, Comm. Algebra 32 (2004), no. 9, 3641 – 3649. · <a href="/1077.20013" class="nowrap">Zbl 1077.20013</a> · <a href="https://doi.org/10.1081/AGB-120039635" class="nowrap">doi:10.1081/AGB-120039635</a></td> </tr><tr> <td>[14]</td> <td class="space">D. L. WHITE, Degree graphs of simple linear and unitary groups, to appear in Comm. Algebra. · <a href="/1105.20007" class="nowrap">Zbl 1105.20007</a></td> </tr><tr> <td>[15]</td> <td class="space">D. L. WHITE, Degree graphs of simple orthogonal and symplectic groups, to appear in J. Algebra. · <a href="/1145.20007" class="nowrap">Zbl 1145.20007</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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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