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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Bender's method</span></span>

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<p>In <a href="Group_theory" title="Group theory">group theory</a>, <b>Bender's method</b> is a method introduced by <a href="#CITEREFBender1970">Bender (1970)</a> for simplifying the local group theoretic analysis of the <a href="Odd_order_theorem" class="mw-redirect" title="Odd order theorem">odd order theorem</a>. Shortly afterwards he used it to simplify the <a href="Walter_theorem" title="Walter theorem">Walter theorem</a> on groups with abelian Sylow 2-subgroups <a href="#CITEREFBender1970b">Bender (1970b)</a>, and Gorenstein and Walter's classification of groups with dihedral Sylow 2-subgroups. Bender's method involves studying a <a href="Maximal_subgroup" title="Maximal subgroup">maximal subgroup</a> <i>M</i> containing the <a href="Centralizer" class="mw-redirect" title="Centralizer">centralizer</a> of an <a href="Involution_(mathematics)#Group_theory" title="Involution (mathematics)">involution</a>, and its <a href="Generalized_Fitting_subgroup" class="mw-redirect" title="Generalized Fitting subgroup">generalized Fitting subgroup</a> <i>F</i><sup>*</sup>(<i>M</i>).

</p><p>One succinct version of Bender's method is the result that if <i>M</i>, <i>N</i> are two distinct maximal subgroups of a simple group with <i>F</i><sup>*</sup>(<i>M</i>) ≤ <i>N</i> and <i>F</i><sup>*</sup>(<i>N</i>) ≤ <i>M</i>, then there is a prime <i>p</i> such that both <i>F</i><sup>*</sup>(<i>M</i>) and <i>F</i><sup>*</sup>(<i>N</i>) are <a href="P-group" title="P-group"><i>p</i>-groups</a>. This situation occurs whenever <i>M</i> and <i>N</i> are distinct maximal parabolic subgroups of a simple group of Lie type, and in this case <i>p</i> is the characteristic, but this has only been used to help identify groups of low Lie rank. These ideas are described in textbook form in <a href="#CITEREFGagen1976">Gagen (1976</a>, p. 43),

<a href="#CITEREFHuppertBlackburn1982">Huppert & Blackburn (1982</a>, Chapter X. 15), <a href="#CITEREFGorensteinLyonsSolomon1996">Gorenstein, Lyons & Solomon (1996</a>, p. 110, Chapter F.19), and <a href="#CITEREFKurzweilStellmacher2004">Kurzweil & Stellmacher (2004</a>, Chapter 10.1).

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBender1970" class="citation cs2">Bender, Helmut (1970), <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.ijm/1256053074">"On the uniqueness theorem"</a>, <i>Illinois Journal of Mathematics</i>, <b>14</b> (3): <span class="nowrap">376–</span>384, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fijm%2F1256053074">10.1215/ijm/1256053074</a></span>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0019-2082">0019-2082</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0262351">0262351</a></cite></li>

<li><cite id="CITEREFBender1970b" class="citation cs2">Bender, Helmut (1970b), "On groups with abelian Sylow 2-subgroups", <i><a href="Mathematische_Zeitschrift" title="Mathematische Zeitschrift">Mathematische Zeitschrift</a></i>, <b>117</b> (<span class="nowrap">1–</span>4): <span class="nowrap">164–</span>176, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01109839">10.1007/BF01109839</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0288180">0288180</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120553015">120553015</a></cite></li>

<li><cite id="CITEREFBenderGlauberman1994" class="citation cs2">Bender, Helmut; <a href="George_Glauberman" title="George Glauberman">Glauberman, George</a> (1994), <i>Local analysis for the odd order theorem</i>, London Mathematical Society Lecture Note Series, vol. 188, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-45716-3</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1311244">1311244</a></cite></li>

<li><cite id="CITEREFGagen1976" class="citation cs2">Gagen, Terence M. (1976), <i>Topics in finite groups</i>, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-21002-7</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0407127">0407127</a></cite></li>

<li><cite id="CITEREFGorensteinLyonsSolomon1996" class="citation cs2"><a href="Daniel_Gorenstein" title="Daniel Gorenstein">Gorenstein, D.</a>; Lyons, Richard; Solomon, Ronald (1996), <a rel="nofollow" class="external text" href="https://www.ams.org/online_bks/surv402"><i>The classification of the finite simple groups. Number 2. Part I. Chapter G</i></a>, Mathematical Surveys and Monographs, vol. 40, Providence, R.I.: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-0390-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1358135">1358135</a></cite></li>

<li><cite id="CITEREFHuppertBlackburn1982" class="citation cs2"><a href="Bertram_Huppert" title="Bertram Huppert">Huppert, Bertram</a>; Blackburn, Norman (1982), <i>Finite groups. III</i>, Grundlehren der Mathematischen Wissenschaften, vol. 243, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-10633-3</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0662826">0662826</a></cite></li>

<li><cite id="CITEREFKurzweilStellmacher2004" class="citation cs2">Kurzweil, Hans; Stellmacher, Bernd (2004), <i>The theory of finite groups</i>, Universitext, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-40510-0</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2014408">2014408</a></cite></li></ul></div><div><div class="zim-footer">

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