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<h1>permut</h1>


<h2>A package to deal with permutability in finite groups</h2>

<p>
    2.0.5</p>

<p>
    11 January 2024
  </p>

</div>
<p><b>
    Adolfo Ballester-Bolinches



  </b>
<br />Email: <span class="URL"><a href="mailto:Adolfo.Ballester@uv.es">Adolfo.Ballester@uv.es</a></span>
<br />Address: <br />Departament de Matemàtiques<br /> Universitat de València<br /> Dr. Moliner, 50<br /> 46100 Burjassot, València, Spain<br />
</p><p><b>
    Enric Cosme-Llópez




  </b>
<br />Email: <span class="URL"><a href="mailto:Enric.Cosme@uv.es">Enric.Cosme@uv.es</a></span>
<br />Homepage: <span class="URL"><a href="https://www.uv.es/coslloen">https://www.uv.es/coslloen</a></span>
<br />Address: <br />Departament de Matemàtiques<br /> Universitat de València<br /> Dr. Moliner, 50<br /> 46100 Burjassot, València, Spain<br />
</p><p><b>
    Ramón Esteban-Romero



  </b>
<br />Email: <span class="URL"><a href="mailto:Ramon.Esteban@uv.es">Ramon.Esteban@uv.es</a></span>
<br />Homepage: <span class="URL"><a href="https://www.uv.es/estebanr">https://www.uv.es/estebanr</a></span>
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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X7A6D71137AE20820">1 <span class="Heading">Introduction to the <strong class="pkg">PERMUT</strong> Package</span></a>
</div>
<div class="ContChap"><a href="chap2_mj.html#X78FA2A627FAD92E6">2 <span class="Heading">Installation and Help of the <strong class="pkg">PERMUT</strong> Package</span></a>
</div>
<div class="ContChap"><a href="chap3_mj.html#X7B87CECD83EA77DF">3 <span class="Heading">Permutability of Subgroups in Finite Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X834D77E17960C0B7">3.1 <span class="Heading">Permutability functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X862727A87C26B10B">3.1-1 ArePermutableSubgroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X875BEB9B7D0DBC9A">3.2 <span class="Heading">Embedding properties related to permutability</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85CFBEBB83245E3A">3.2-1 PermutMaxTries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80C15D427EC11A5F">3.2-2 IsPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8244918D7D1C2703">3.2-3 OneSubgroupNotPermutingWith</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X865C7F81823FFA10">3.2-4 IsFPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X826D28757824F83E">3.2-5 OneFSubgroupNotPermutingWith</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7BA6690983912C09">3.2-6 IsSPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83E103AF7C3DC9B7">3.2-7 OneSylowSubgroupNotPermutingWith</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86A62A52831BE00E">3.2-8 IsSNPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83D1BF5778D6790A">3.2-9 OneSystemNormaliserNotPermutingWith</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B2E052E7F9C4251">3.2-10 IsConjugatePermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X809E1C8E8384ADE8">3.2-11 OneConjugateSubgroupNotPermutingWith</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AC58CFF87770C6C">3.2-12 IsWeaklySPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X870923B4798E5097">3.2-13 OneElementShowingNotWeaklySPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85E85FBF78DAB08E">3.2-14 IsWeaklyPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8427C8DD78732D56">3.2-15 OneElementShowingNotWeaklyPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84BE89EF8140D2C0">3.2-16 IsWeaklyNormal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X82C8AE027E578ACF">3.2-17 OneElementShowingNotWeaklyNormal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X872EA72A7CEC3E65">3.2-18 IsWithSubnormalizerCondition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7FBEC9DD7C8AED52">3.2-19 OneSubgroupInWhichSubnormalNotNormal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87CA542C7E1FEEF3">3.2-20 IsWithSubpermutizerCondition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79648B4F8758E187">3.2-21 OneSubgroupInWhichSubnormalNotPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X831F5B8787BB111F">3.2-22 IsWithSSubpermutizerCondition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AC9002B7D5AB2DA">3.2-23 OneSubgroupInWhichSubnormalNotSPermutable</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4_mj.html#X7E7EFDD1878F599D">4 <span class="Heading">T-groups, PT-groups, and PST-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7ABC289380DB85BE">4.1 <span class="Heading">"One" functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7E15C5F778535A28">4.1-1 OneSubnormalNonNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F5EE21F85A8D14F">4.1-2 OneSubnormalNonPermutableSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7D7717657B9C7BEB">4.1-3 OneSubnormalNonSPermutableSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7878496C869CB90C">4.1-4 OneSubnormalNonConjugatePermutableSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B23E3857CA178FE">4.1-5 OneSubnormalNonSNPermutableSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7B13DA0A800D8635">4.2 <span class="Heading">Group properties related to permutability</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7930E5AD817BE4B3">4.2-1 IsTGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7CF2634F84EC572B">4.2-2 IsPTGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7BB4D5A782FF5B0E">4.2-3 IsPSTGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8789176282320022">4.2-4 IsCPTGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B9EBEA07DBE246F">4.2-5 IsPSNTGroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5_mj.html#X83F7986D84F33F6C">5 <span class="Heading">Local Functions in the <strong class="pkg">PERMUT</strong> Package</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7E8389FD7DE9769C">5.1 <span class="Heading">A Local Function for Supersolubility</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7B9F19A37F4A5B35">5.1-1 IsPSupersolvable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7C4B2A9A7F2D7E0B">5.2 <span class="Heading">Local functions for T-groups, PT-groups, and PST-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X780657F179FFB7EE">5.2-1 IsCp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X82953B4878F88327">5.2-2 IsXp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8304885C78E4A584">5.2-3 IsYp</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X8540D4B679519F8B">5.3 <span class="Heading">Auxiliary Functions for T-groups, PT-groups, and PST-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8775CAFD80D3FC02">5.3-1 IsAbCp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7A6372D3871E8E55">5.3-2 IsDedekindSylow</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X812C3A017F534CC0">5.3-3 IwasawaTripleWithSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7E86B21086879C48">5.3-4 IwasawaTriple</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8121A92F7EABEA0C">5.3-5 IsIwasawaSylow</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6_mj.html#X82E82E45786E1B22">6 <span class="Heading">Totally and Mutually Permutable Products</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X85291E9D86736F21">6.1 <span class="Heading">Functions for Mutually and Totally Permutable Products</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X876F4E0D7C796134">6.1-1 AreMutuallyPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X80225ABC79C7875F">6.1-2 OnePairShowingNotMutuallyPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7D58DFE57AA4573B">6.1-3 AreTotallyPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7B51474D878ACD11">6.1-4 OnePairShowingNotTotallyPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X823C6D68813B3AC8">6.1-5 AreMutuallyFPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X83E0517A800AB839">6.1-6 OnePairShowingNotMutuallyFPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7B9DA1C2834CA53D">6.1-7 AreTotallyFPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7CD0B5737A25C1AD">6.1-8 OnePairShowingNotTotallyFPermutableSubgroups</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap7_mj.html#X840872AB823EB3DB">7 <span class="Heading">Other Functions in the <strong class="pkg">PERMUT</strong> Package</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X86FA580F8055B274">7.1 <span class="Heading">Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X81EA23787BCD8633">7.1-1 AllSubnormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X858C07DA815A2E9A">7.1-2 PrimesDividingSize</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7B07AAD37F7CEDEB">7.1-3 SylowSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7E01D1C47814DA3C">7.1-4 IsSCGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X798736C17DD23F8A">7.1-5 IsSylowTowerGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X81FCDD7C7A5D1658">7.1-6 Permutizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7D47DF77826C1BE0">7.1-7 AllGeneratorsCyclicPGroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
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