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<p><a id="X84CA5C9B81900889" name="X84CA5C9B81900889"></a></p>
<div class="ChapSects"><a href="chap2.html#X84CA5C9B81900889">2 <span class="Heading">Basic functionality for <span class="SimpleMath">ZG</span>-resolutions and group cohomology</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7C0B125E7D5415B4">2.1 <span class="Heading"> Resolutions</span></a>
<span
<div class="ContSSBlock">
<span class="ContSS
<span class="ContSS"><br /><span class="nocss">  >< class="utton">Examples:</strong<span class"RL"<a href"./.html">1</a></>  <pan =URL< ="..//chap10.>2a/ ,<"a=.www//aboutPoincareSeries"6/pan  span class=">ahref///."<>> span =">a="../ww///aboutExtensions.html>/panspan class=""a =./wwSideLinksAbout/boutTorAndExt.html>/>
<span class="ContSS"><br /><span class="nocss"> 
s class""><br/< class""&;nbsp< =.#">214 ResolutionCubicalCrystGroup/pan>
<span class="ContSS"><br /><
<span class<< ="X79C31EED8406A3E9 =X79C31EED8406A3E9>/>/p
<span class="ContSS"><br /><span
<<h5>2<
<span class="ContSS"><
<span class classfunc"tableclass=func"width0><tr ="tdleft>code=">&82;</>(var="">A/var var="&;functionnbsp)/d>/>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79A0221B7E96B642">2.1-11 ResolutionSubgroup</a></span>
</div/div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span
<span
<div class="ContSSBlock">
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
/>d>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7F9E1F1781479F7B">2.3 <span class="Heading"> Resolutions <span class="SimpleMath">⟶</span> (Co)chain Complexes</span></a>
</>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="
< classContSS> class>nbsp>< =".htmlX81FED0E9858E413A>.-2 HomToIntegralModule<>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X83BA99787CBE2B7D">2.3-3 TensorWithIntegers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><
<d></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X80B6849C835B7F19">2.4 <span class="Heading"> Cohomology
</span>
<div class ="">table=func=10"<>">< ="">&82;HilbertPoincareSeries>(varArg> <>< class(&;function)td></></>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79C31EED8406A3E9">2.4-1 AreIsomorphicGradedAlgebras</a></span>
<span class="ContSS"><r />spanclassnocss&; /><a href=".tmlX3>.-2HAPDerivation
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B93B7D082A50E61">2.4java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span
<span class="ContSS"><br /><span class="nocss"> >aid=X803D9B5E7A26F749""<<p>
<spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< class< /< classnbsp;span =".htmlX828D20AC8735152B".7LHSSpectralSequenceLastSheet
<span class="ContSS"><br /><span class="nocss">  <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< class>br/>< class"nbsp; < ="chap2htmlX796632C585D47245"2.49 ModPCohomologyRing
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X831034A284F3906F">2.4-10 Mod2CohomologyRingPresentation</a></span>
</div></div>
<div <p>Inputs a <span="impleMath"d   <span a quotientclassSimpleMath"ER=[.x_m<>overafield F/>. It alist classSimpleMath>[,,]/>  span=SimpleMath<is polynomial and ="">J/>  alist generators    spanSimpleMath></> such there  isomorphism =""α /→ d ~ d<>. isomorphism to  homomorphism classSimpleMathh S→ker<span  function
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D1658EF810022E5">2.5-1 GroupCohomology</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7F0A19E97980FD57">2.5-2 GroupHomology</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7A30C1CC7FB6B2E9">2.5-3 PrimePartDerivedFunctor</a></span<p>strong="button">xamples:/></
<span class="ContSS"><<>aidX855D2D747B6C54E1=X855D2D747B6C54E1/><>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X828B81D9829328F8">2.5-5 PoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7EFE814686C4EEF5">2.5-6 RankHomologyPGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86CDD4B77CBE3087">2.6 <span class="Heading"> <span class="SimpleMath">F_p</span>-modules</span></a>
<span
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X85758F95832207D2">2.6-1 GroupAlgebraAsFpGModule</a></span>
span="ContSS"><br /span=""&; ><ahref.#>.-Radicala<span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7929281B848A9FBE">2.6-3 RadicalSeries<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</div></div>
</div>

<h3>2 <span class="Heading">Basic functionality for <span class="SimpleMath">ZG</span>-resolutions and group cohomology</span></h3>

<p>This page coversh524-6LHSSpectralSequenceh5

<p><a id="X7C0B125E7D5415B4" name="X7C0B125E7D5415B4"></a></p>

<h4>2.1

<p><a id="X868E2A04832619C5" name="X868E2A04832619C5"></a></p>

<h5>2.1-1 EquivariantChainMap</h5><divclassfunc><table classfunc="100"<><tdclass"> class"">RLHSSpectralSequence">">">><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivariantChainMap</code>( <var class="Arg">R</var>, <var class="Arg">S</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> of <span class="SimpleMath">Z</span>, a free <span class="SimpleMath">ZQ</span>-resolution <span class="SimpleMath">S</span> of <span class="SimpleMath">Z</span>, and a group homomorphism <span class="SimpleMath">f: G → Q</span>. It returns the induced <span class="SimpleMath">f</span>-equivariant chain map <span class="SimpleMath">F: R → S</span>.</p>

<p><strong>247LHSSpectralSequenceLastSheeth5>

<p><a id="X79EA11238403019D" name="X79EA11238403019D"></a></p>

<h5>2.1-2 FreeGResolution</h5>

<div class="unc"t "func" width"00"<><td="">#27;LHSSpectralSequenceLastSheet> var""><var class"N/>)/>td =tdright"(nbspfunctionnbsptdtr<table<div
 non-free span=><>spanSimpleMath</>  positive span=S><.It toreturnspanclass"/> termsofafreespanclass"">ZG/span>-resolution of SimpleMath>/>   groupsrbe     resolutionswithhomotopiesthemp

<p>The contracting homotopy on the resolution was implemented by Bui Anh Tuan.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap6.html">2</a></span> , <span class="URL"><a href="../tutorial/chap7.html">3</a></span> , <span class="URL"><a href="../tutorial/chap11.html">4</a></span> , <span class="URL"><a href="../tutorial/chap13.html">5</a></span> , <span class="URL"><a href="../tutorial/chap14.html">6</a></span>

<p><a id="X7CA87AA478007468" name="X7CA87AA478007468"<h52.-8ModPCohomologyGeneratorsh5>

<h5>2.1-3 ResolutionBieberbachGroup</h5>

<div class="func">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>< <td"dright"&;function;/d<tr>/>
<p>Inputs a torsion free crystallographic group <span class="SimpleMath">G</span>, also known as a Bieberbach group, represented using <strong class="button">AffineCrystGroupOnRight</strong> as in the GAP package Cryst. It also optionally inputs a choice of vector <span class="SimpleMath">v</span> in the Euclidean space <span class="SimpleMath">R^n</span> on which <span class="SimpleMath">G</span> acts freely. The function returns <span class="SimpleMath">n+1</ =functable=f"width=0%>tr>
codeclass"func">&82;ModPCohomologyGenerators( Arg">R)/td>(n;&bsptd>table/iv

<p>This function is part of the HAPcryst package written by Marc Roeder and thus requires the HAPcryst package to be loaded.</p>

<p>The function requires the use of Polymake software.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> </p>

<p><a id="X81A5CEFC82A1897D" name="X81A5CEFC82A1897D"></a></p>

<h5>2.1-4 ResolutionCubicalCrystGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td
<p>Inputs a<p><strong c="button">Examples</strong> <span="URL"><ahref="../www/SideLinks/About/aboutIntro.html"1<a<span/>

<p>This function was written by Bui Anh java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URLjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

>=""="<>/>

<h5>2.1-5 ResolutionFiniteGroup</h5>

<div class="func">table="func ="0%>< ""< ="> ModPCohomologyRing( var class=Arg"R<var="td="">&;function&;)
<p>< classfunc class"=0"trtdclass=""><codeclass=func87 ModPCohomologyRingcode(< class="Arg">G/>,< classArg<var</tdtd="">(nbsp t>/r<

<p><strong class="button">Examples:</strong> <span class"URL"><a="..//.html><href/html2</a><span  span=URL< =".tutorialchap8"></>/span,span=URL><a href./."<>span>  span class=URL"><a href.tutorial>5</>/>  span="">a="./SideLinks/bout/aboutParallelhtml>, < "ahref"..//SideLinksA/.>< href".www/About/aboutCocycles"<>/span  RLa href"./www//AboutaboutPeriodic.html"9</a></> , span=""><a href"..wwwSideLinks/About/aboutCohomologyRings.html">10<a><span,<span classURL">1/span , URL> ="../www//About/.html"></<> ,<span ="URL"<a href"./www/SideLinks/About/aboutDefinitions.html">13</a></span> , <span class="URL">a href="./www/SideLinks//.">4/>/span  <spanclass""<href"../www/SideLinks/About/boutExtensions.>1/span  < =.///boutaboutGouterhtml>18/a>,URL =./SideLinksAbout.">9aURL">a href"./www/About.">0

<p><a id="X7CBE6BDA7DB5AD7D" name="X7CBE6BDA7DB5AD7D"></a></p>

<h5>2.1-6 ResolutionNilpotentGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionNilpotentGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
p anilpotent < =""</span can )  an< classk≥>  returnsclass>1<span of freespan"">ZGspan of=""Z/pan>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCohomologyRings.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRosenbergerMonster.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">5</a></span> </p>

<p>< id"" name=">/

<h57ResolutionNormalSeries>

<div class="func"><table class="func" width="100%"><tr><td
<p>Inputs>241 </h5

<p><strong class="button">Examples:</strong> <span class=java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0

<p><a id="X86934BE9858F7199" name="X86934BE9858F7199"></a></p>

<h5>2.1-8 ResolutionPrimePowerGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">̶<;iv="func"><table="func"width="10%"<><td ="tdleft">codeclass">̻ Mod2CohomologyRingPresentation( Atd class="tdright>nbsp&;<td>/>/>
<p>Inputsfinite class"SimpleMath">p/>-group<spanclassSimpleMathG/span>and integer span="SimpleMath">k ≥1</> Itreturnsspan=SimpleMath>k+<span terms a minimal <pan="impleMath"FG/span- of the <span class"SimpleMath">F</span <spanclass=SimpleMath><span elementspjava.lang.StringIndexOutOfBoundsException: Index 361 out of bounds for length 361

<p><strong class="button">Examples:</strong   classSimpleMath2span <span ""<spanthisfunction   forthe< =">">H^(G )>.TheLyndon-Hochschild-Serre spectral is prove complete thefunction applied a< =""2/>-group positive ="impleMathn<>  first <=SimpleMath>+</> ofa  <class""<span < =SimpleMathR/,constructs   algebra<pan=SimpleMathAH∗ ( <span    < class"Aapproximate for"H*(,F)span For" large < =SimpleMath<spanapproximationbepresentation< ">∗,)/span> Alternatively, beapplied directlytoeithertheresolution

<p><a id="X7E4556B078B209CE" name="X7E4556B078B209CE"></a></p>

<h5>2.1-9 ResolutionSL2Z</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSL2Z</code>( <var class="Arg">m</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs positive integers <span class="SimpleMath">m, n</span> and returns <span class="SimpleMath">n</span> terms of a free <span class="SimpleMath">ZG</span>-resolution of <span class="SimpleMath">Z</span> for the group <span class="SimpleMath">G=SL_2( Z[1/m])</span>.</p>

<p>This function is joint work with Bui Anh Tuan.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../tutorial/chap13.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">4</a></span> </p>

<p><a id="X8518446086A3F7EA" name="X8518446086A3F7EA"></a></p>

<h5>2.1-10 ResolutionSmallGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSmallGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSmallGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Inputs<>< ="X7BCF8D907D237A03 name=X7BCF8D907D237A03>java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59

<p>If <span class="SimpleMath">G</span> is a

<> function written Irina.<p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> </p>

<p><a id="X79A0221B7E96B642" name="X79A0221B7E96B642"></a></p>

<h5>2.1-11 ResolutionSubgroup</h5>

<divclass"< class""width"0"><code="func>&27 ResolutionSubgroup/> var =Arg"R<varvar="Arg">H</> )</td<tdclasstdright>(nbsp )<tdtrtablediv
<>Inputs freeZG-resolution  < class=SimpleMath>Z<span afinite span=SimpleMath≤<span   free ZH-resolution of =SimpleMath>Z</span./pjava.lang.StringIndexOutOfBoundsException: Index 208 out of bounds for length 208

p< class=">Examples:< class="><ahref.////."1/a , ">< href=.www/About.html"2/>/> , < classURLhrefwww/java.lang.StringIndexOutOfBoundsException: Range [295, 294) out of bounds for length 318

<p><a id="X85EC9D8E7A15A570" name="X85EC9D8E7A15A570"></a></p>

<h4>2.2 <span class="Heading"> Algebrasp< classbuttonExamplesstrong <span class="URL"><a href"../tutorial/chap7.html">1</a></span , <span class""><a href="./tutorial/chap8.html">2</><span<p>

<< "" name="X7D5DD19D7BA9D816"></>/>

<h5>2.2-1 LeibnizComplex</h5>

<div class="func"><table class="func" width="100%"><tr><h5>.5- GroupHomologyh5
<p>Inputs

<p><strong class="button">Examples classfunc>table="func" width10"<dleft< class"">&82;GroupHomology>(< class"">G<varvar="Arg"k/> )<td class="tdright>&;function&;)

<p><a id="X7F9E1F1781479F7B" name="X7F9E1F1781479F7B"></a></p>

>23span=""> < class"⟶Cochain/><>

<p><a id="X788F3B5E7810E309" name="X788F3B5E7810E309"></a></p>

<h5>2.3-1 HomToIntegers</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToIntegers</code>(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToIntegers</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
< class"< class"" ="10")"&;function )/>/>/>/iv>
<p>Inputs a chain complex <span class="SimpleMath">C</span> of free abelian groups and returns the cochain complex <span class="SimpleMath">Hom_ Z(C, Z)</span>.</p>

<p>Inputs a free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> in characteristic <span class="SimpleMath">0</span> and returns the cochain complex <span class="SimpleMath">Hom_ ZG(R, Z)</span>.</p>

<p>Inputs an equivariant chain map <span class="SimpleMath">F: R→ S</span> of resolutions and returns the induced cochain map <span class="SimpleMath

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X81FED0E9858E413A" name="X81FED0E9858E413A"></a></p>

<h5>2.3-2 HomToIntegralModule</h5>

< class="func width="0%><tr classtdleftcodeclass="&8HomToIntegralModule varclass"">R, var ="Arg<vartd ="dright"> function )</td>/></></div
<p>Inputs a free <span class="SimpleMath">ZGjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap13.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTwistedCoefficients.html">3</a></span> </p>

<p><a id="X83BA99787CBE2B7D" name="X83BA99787CBE2B7D"></a></p>

<h5>2.3-3 TensorWithIntegers

class ="func"width0>>< classcode=func87 </>(  class<var</td>td=tdright;function)/d>/><>/div
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegers</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
h">Rofcharacteristic SimpleMath<>     <">R _ ZGZ/>.



<p>Inputs an equivariant chain map <span class="SimpleMath">F: R → S</span> in characteristic <span class="SimpleMath">0</span> and returns the div=">table = ="0>>< classcode=""&27 />< =ArgL<> <var""n/>)td"">&;function/>/><table</div

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="."n/>.  returnsa ofpolynomials < classSimpleMath>(=x/Q()span  expansion has coefficient of <span class=SimpleMath>^/>equalthe  vector space< =SimpleMath>(,F_p>   spanclass""><>  range< classSimpleMath ≤ ≤n<span.( secondinput   ,     trieschoose' . - thefunction classbuttonP()/> be used an< =SimpleMath>() can input a < =">nspanterms)a minimal< =""><s>  =>/>for ="SimpleMath"></span Alternatively    variable   =>> integersIn case coefficient <span="^><span  equaljava.lang.StringIndexOutOfBoundsException: Index 1363 out of bounds for length 1363

<p><a id="X8122D25786C83565" name="X8122D25786C83565"></a></p>

<h5>2.3-4 TensorWithIntegersModP</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegersModP</code>( <var class="Arg">C</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegersModP</code>( <var class="Arg">R</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegersModP</code>( <var class="Arg">F</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a chain complex <span class="SimpleMath">C</span> of characteristic <span class="SimpleMath">0</span> and a prime integer <span class="SimpleMath">p</span>. It returns the chain complex <span class="SimpleMath">C ⊗_ Z Z_p</span> of characteristic <span class="SimpleMath">p</span>.</p>

<p>Inputs a free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> of characteristic <span class="SimpleMath">0</span> and a prime integer <span class="SimpleMath">p</span>. It returns the chain complex <span class="SimpleMath">R ⊗_ ZG Z_p</span> of characteristic <span class="SimpleMath">p</span>.</p>

<p>Inputs an equivariant chain map <span class="SimpleMath">F: R → S</span> in characteristic <span class="SimpleMath">0</span> a prime integer <span class="SimpleMath">p</span>. It returns the induced chain map <span class="SimpleMath">F⊗_ ZG Z_p : R ⊗_ ZG Z_p ⟶ S ⊗_ ZG Z_p</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPerformance.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeries.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutDefinitions.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTorAndExt.html">9</a></span> </p>

<p><a id="X80B6849C835B7F19" name="X80B6849C835B7F19"></a></p>

<h4>2.4 <span class="Heading"> Cohomology rings</span></h4>

<p><a id="X79C31EED8406A3E9" name="X79C31EED8406A3E9"></a></p>

<h5>2.4-1 AreIsomorphicGradedAlgebras</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AreIsomorphicGradedAlgebras</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two freely presented graded algebras <span class="SimpleMath">A= F[x_1, ..., x_m]/I</span> and <span class="SimpleMath">B= F[y_1, ..., y_n]/J</span> and returns <strong class="button">true</strong> if they are isomorphic, and <strong class="button">false</strong> otherwise. This function was implemented by Paul Smith.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X83DC2F1A805BA7A3" name="X83DC2F1A805BA7A3"></a></p>

<h5>2.4-2 HAPDerivation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HAPDerivation</code>( <var class="Arg">R</var>, <var class="Arg">I</var>, <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a polynomial ring <span class="SimpleMath">R= F[x_1,...,x_m]</span> over a field <span class="SimpleMath">F</span> together with a list <span class="SimpleMath">I</span> of generators for an ideal in <span class="SimpleMath">R</span> and a list <span class="SimpleMath">L=[y_1,...,y_m]⊂ R</span>. It returns the derivation <span class="SimpleMath">d: E → E</span> for <span class="SimpleMath">E=R/I</span> defined by <span class="SimpleMath">d(x_i)=y_i</span>. This function was written by Paul Smith. It uses the Singular commutative algebra package.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7B93B7D082A50E61" name="X7B93B7D082A50E61"></a></p>

<h5>2.4-3 HilbertPoincareSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HilbertPoincareSeries</code>( <var class="Arg">E</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a presentation <span class="SimpleMath">E= F[x_1,...,x_m]/I</span> of a graded algebra and returns its Hilbert–Poincaré series. This function was written by Paul Smith and uses the Singular commutative algebra package. It is essentially a wrapper for Singular's Hilbert–Poincaré series.



<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap8.html">1</a></span> </p>

<p><a id="X803D9B5E7A26F749" name="X803D9B5E7A26F749"></a></p>

<h5>2.4-4 HomologyOfDerivation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomologyOfDerivation</code>( <var class="Arg">d</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a derivation <span class="SimpleMath">d: E → E</span> on a quotient <span class="SimpleMath">E=R/I</span> of a polynomial ring <span class="SimpleMath">R= F[x_1,...,x_m]</span> over a field <span class="SimpleMath">F</span>. It returns a list <span class="SimpleMath">[S,J,h]</span> where <span class="SimpleMath">S</span> is a polynomial ring and <span class="SimpleMath">J</span> is a list of generators for an ideal in <span class="SimpleMath">S</span> such that there is an isomorphism <span class="SimpleMath">α: S/J → ker d/ im~ d</span>. This isomorphism lifts to the ring homomorphism <span class="SimpleMath">h: S → ker d</span>. This function was written by Paul Smith. It uses the Singular commutative algebra package.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X855D2D747B6C54E1" name="X855D2D747B6C54E1"></a></p>

<h5>2.4-5 IntegralCohomologyGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntegralCohomologyGenerators</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs at least <span class="SimpleMath">n+1</span> terms of a free <span class="SimpleMath">ZG</span>-resolution of <span class="SimpleMath">Z</span> and the integer <span class="SimpleMath">n ≥ 1</span>. It returns a minimal list of cohomology classes in <span class="SimpleMath">H^n(G, Z)</span> which, together with all cup products of lower degree classes, generate the group <span class="SimpleMath">H^n(G, Z)</span> . (Let <span class="SimpleMath">a_i</span> be the <span class="SimpleMath">i</span>-th canonical generator of the <span class="SimpleMath">d</span>-generator abelian group <span class="SimpleMath">H^n(G,Z)</span>. The cohomology class <span class="SimpleMath">n_1a_1 + ... +n_da_d</span> is represented by the integer vector <span class="SimpleMath">u=(n_1, ..., n_d)</span>. )</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7F5D00C97A46D686" name="X7F5D00C97A46D686"></a></p>

<h5>2.4-6 LHSSpectralSequence</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LHSSpectralSequence</code>( <var class="Arg">G</var>, <var class="Arg">N</var>, <var class="Arg">r</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">2</span>-group <span class="SimpleMath">G</span>, and normal subgroup <span class="SimpleMath">N</span> and an integer <span class="SimpleMath">r</span>. It returns a list of length <span class="SimpleMath">r</span> whose <span class="SimpleMath">i</span>-th term is a presentation for the <span class="SimpleMath">i</span>-th page of the Lyndon-Hochschild-Serre spectral sequence. This function was written by Paul Smith. It uses the Singular commutative algebra package.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X828D20AC8735152B" name="X828D20AC8735152B"></a></p>

<h5>2.4-7 LHSSpectralSequenceLastSheet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LHSSpectralSequenceLastSheet</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">2</span>-group <span class="SimpleMath">G</span> and normal subgroup <span class="SimpleMath">N</span>. It returns presentation for the <span class="SimpleMath">E_∞</span> page of the Lyndon-Hochschild-Serre spectral sequence. This function was written by Paul Smith. It uses the Singular commutative algebra package.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7DEFADD17CAA6308" name="X7DEFADD17CAA6308"></a></p>

<h5>2.4-8 ModPCohomologyGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyGenerators</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyGenerators</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and positive integer <span class="SimpleMath">n</span>, or else <span class="SimpleMath">n+1</span> terms of a minimal <span class="SimpleMath">FG</span>-resolution <span class="SimpleMath">R</span> of the field <span class="SimpleMath">F</span> of <span class="SimpleMath">p</span> elements. It returns a pair whose first entry is a minimal list of homogeneous generators for the cohomology ring <span class="SimpleMath">A=H^∗(G, F)</span> modulo all elements in degree greater than <span class="SimpleMath">n</span>. The second entry of the pair is a function <strong class="button">deg</strong> which, when applied to a minimal generator, yields its degree. WARNING: the following rule must be applied when multiplying generators <span class="SimpleMath">x_i</span> together. Only products of the form <span class="SimpleMath">x_1*(x_2*(x_3*(x_4*...)))</span> with <span class="SimpleMath">deg(x_i) ≤ deg(x_i+1)</span> should be computed (since the <span class="SimpleMath">x_i</span> belong to a structure constant algebra with only a partially defined structure constants table).</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">1</a></span> </p>

<p><a id="X796632C585D47245" name="X796632C585D47245"></a></p>

<h5>2.4-9 ModPCohomologyRing</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyRing</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyRing</code>( <var class="Arg">R</var>, <var class="Arg">level</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyRing</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyRing</code>( <var class="Arg">G</var>, <var class="Arg">n</var>, <var class="Arg">level</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and positive integer <span class="SimpleMath">n</span>, or else <span class="SimpleMath">n</span> terms of a minimal <span class="SimpleMath">FG</span>-resolution <span class="SimpleMath">R</span> of the field <span class="SimpleMath">F</span> of <span class="SimpleMath">p</span> elements. It returns the cohomology ring <span class="SimpleMath">A=H^∗(G, F)</span> modulo all elements in degree greater than <span class="SimpleMath">n</span>. The ring is returned as a structure constant algebra <span class="SimpleMath">A</span>. The ring <span class="SimpleMath">A</span> is graded. It has a component <strong class="button">A!.degree(x)</strong> which is a function returning the degree of each (homogeneous) element <span class="SimpleMath">x</span> in <strong class="button">GeneratorsOfAlgebra(A)</strong>. An optional input variable <span class="SimpleMath">"level"</span> can be set to one of the strings <span class="SimpleMath">"medium"</span> or <span class="SimpleMath">"high"</span>. These settings determine parameters in the algorithm. The default setting is <span class="SimpleMath">"medium"</span>. When <span class="SimpleMath">"level"</span> is set to <span class="SimpleMath">"high"</span> the ring <span class="SimpleMath">A</span> is returned with a component <strong class="button">A!.niceBasis</strong>. This component is a pair <span class="SimpleMath">[Coeff,Bas]</span>. Here <span class="SimpleMath">Bas</span> is a list of integer lists; a "nice" basis for the vector space <span class="SimpleMath">A</span> can be constructed using the command <strong class="button">List(Bas,x->Product(List(x,i->Basis(A)[i]))</strong>. The coefficients of the canonical basis element <strong class="button">Basis(A)[i]</strong> are stored as <strong class="button">Coeff[i]</strong>. If the ring <span class="SimpleMath">A</span> is computed using the setting <span class="SimpleMath">"level"="medium"</span> then the component <strong class="button">A!.niceBasis</strong> can be added to <span class="SimpleMath">A</span> using the command <strong class="button">A:=ModPCohomologyRing_part_2(A)</strong>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap8.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutModPRings.html">2</a></span> </p>

<p><a id="X831034A284F3906F" name="X831034A284F3906F"></a></p>

<h5>2.4-10 Mod2CohomologyRingPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
p applied afinite< classSimpleMath2/span- < class"< returnsa for 2/span cohomologyring H^(G,F)</>. The spectralsequenceisusedtoprove thepresentationiscomplete  thefunctionisappliedtoa<span ="">2/span>-roupGand positive <spanclass"SimpleMath">n/span  function first  <spanclass=SimpleMath>+1/spantermsofafree<panclass"SimpleMath"><span-resolution< ="impleMath"></span> thenconstructs  algebra< ="">AH(∗≤ )G )/> andfinally spanclass=""A<span to  a presentationfor< class"impleMath">^*(G F)<span.For" large" <spanclass=SimpleMath>n/span theapproximationwillbe a correctforclass">^(G F)/span. Alternatively, the function can be applied directly toeitherthe resolution ">R/> or algebra SimpleMath">Aspan.Thisfunction was written by Paul Smith. It uses theSingularcommutativealgebrapackageto the Lyndon-Hochschild-Serre spectralsequence.



<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap8.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">2</a></span> </p>

<p><a id="X7BCF8D907D237A03" name="X7BCF8D907D237A03"></a></p>

<h4>2.5 <span class="Heading"> Group Invariants</span></h4>

<p><a id="X7D1658EF810022E5" name="X7D1658EF810022E5"></a></p>

<h5>2.5-1 GroupCohomology</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupCohomology</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupCohomology</code>( <varjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Inputs a group <span class="SimpleMath"><div classfunctable="func ="0"
/tr>
/divjava.lang.StringIndexOutOfBoundsException: Index 277 out of bounds for length 277

<p>If a prime <span class="SimpleMath">p</span> is given as an optional third input variable then the function returns the list of abelian invariants of <span class="SimpleMath">H^k(G, Z_p)</span>. In this case each abelian invariant will be equal to <span class="SimpleMath">p</span> and the length of the list will be the dimension of the vector space <span class="SimpleMath">H^k(G, Z_p)</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class=>a id =""><a<pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<p><a id="X7F0A19E97980FD57" name="X7F0A19E97980FD57"></a></p<p> ="X85758F95832207D2 =X85758F95832207D2>/>/>

<h5- </>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupHomology</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupHomology</code>( <var class="Arg">G</var>, <var class="Arg">k</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">G</span> and integer <span class

<

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap13.html">2</a>

<p><a id="X7A30C1CC7FB6B2E9" name="X7A30C1CC7FB6B2E9"></a></p>

<h5>2.5-3 PrimePartDerivedFunctor</h5>

">‣PrimePartDerivedFunctor/code( Arg"G/>, R, >k/var><td="tdright">&bsp </d>/></tablediv>
<p>Inputs a group <span class="SimpleMath">G</span>, an integer <span class="SimpleMath">k ≥ 0</span>, at least <span class="SimpleMath">k+1</span> terms of a free <span class="SimpleMath">ZP</span>-resolution

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> ,a=""name<<java.lang.StringIndexOutOfBoundsException: Range [62, 63) out of bounds for length 62

<p><a id="X828B81D9829328F8" name="X828B81D9829328F8"></a></p>

<h5>2.5-4 PoincareSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">L</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and a positive integer <span class="SimpleMath">n</span>. It returns a quotient of polynomials <span class="SimpleMath">f(x)=P(x)/Q(x)</span> whose expansion has coefficient of <span class="SimpleMath">x^k</span> equal to the rank of the vector space <span class="SimpleMath">H_k(G, F_p)</span> for all <span class="SimpleMath">k</span> in the range <span class="SimpleMath">1 ≤ k ≤ n</span>. (The second input variable can be omitted, in which case the function tries to choose a `reasonable' value for n. For 2-groups the function PoincareSeriesLHS(G) can be used to produce an f(x) that is correct in all degrees.) In place of the group G the function can also input (at least n terms of) a minimal mod-p resolution R for G. Alternatively, the first input variable can be a list L of integers. In this case the coefficient of x^k in f(x) is equal to the (k+1)st term in the list.



<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap8.html">2</a></span> , <span class="URL"><a href="../tutorial/chap11.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutModPRings.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeries.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeriesII.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTorAndExt.html">

<p><a id="X828B81D9829328F8" name="X828B81D9829328F8"></a></p>

<h5>2.5-5 PoincareSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">L</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and a positive integer <span class="SimpleMath">n</span>. It returns a quotient of polynomials <span class="SimpleMath">f(x)=P(x)/Q(x)</span> whose expansion has coefficient of <span class="SimpleMath">x^k</span> equal to the rank of the vector space <span class="SimpleMath">H_k(G, F_p)</span> for all <span class="SimpleMath">k</span> in the range <span class="SimpleMath">1 ≤ k ≤ n</span>. (The second input variable can be omitted, in which case the function tries to choose a `reasonable' value for n. For 2-groups the function PoincareSeriesLHS(G) can be used to produce an f(x) that is correct in all degrees.) In place of the group G the function can also input (at least n terms of) a minimal mod-p resolution R for G. Alternatively, the first input variable can be a list L of integers. In this case the coefficient of x^k in f(x) is equal to the (k+1)st term in the list.



<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap8.html">2</a></span> , <span class="URL"><a href="../tutorial/chap11.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutModPRings.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeries.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeriesII.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTorAndExt.html">9</a></span> </p>

<p><a id="X7EFE814686C4EEF5" name="X7EFE814686C4EEF5"></a></p>

<h5>2.5-6 RankHomologyPGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RankHomologyPGroup</code>( <var class="Arg">G</var>, <var class="Arg">P</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span>, a rational function <span class="SimpleMath">P</span> representing the Poincaré series of the mod-<span class="SimpleMath">p</span> cohomology of <span class="SimpleMath">G</span> and a positive integer <span class="SimpleMath">n</span>. It returns the minimum number of generators for the finite abelian <span class="SimpleMath">p</span>-group <span class="SimpleMath">H_n(G, Z)</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> </p>

<p><a id="X86CDD4B77CBE3087" name="X86CDD4B77CBE3087"></a></p>

<h4>2.6 <span class="Heading"> <span class="SimpleMath">F_p</span>-modules</span></h4>

<p><a id="X85758F95832207D2" name="X85758F95832207D2"></a></p>

<h5>2.6-1 GroupAlgebraAsFpGModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupAlgebraAsFpGModule</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and returns the modular group algebra <span class="SimpleMath">F_pG</span> in the form of an <span class="SimpleMath">F_pG</span>-module.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X84B5182E831D0928" name="X84B5182E831D0928"></a></p>

<h5>2.6-2 Radical</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Radical</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">F_pG</span>-module and returns its radical.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7929281B848A9FBE" name="X7929281B848A9FBE"></a></p>

<h5>2.6-3 RadicalSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RadicalSeries</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RadicalSeries</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">F_pG</span>-module <span class="SimpleMath">M</span> and returns its radical series as a list of <span class="SimpleMath">F_pG</span>-modules.</p>

<p>Inputs a free <span class="SimpleMath">F_pG</span>-resolution R and returns the filtered chain complex <span class="SimpleMath">⋯ Rad_2( F_pG)R ≤ Rad_1( F_pG)R ≤ R</span>.</p>

<p><strong class="button">Examples:</strong></p>


<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap1.html">[Previous Chapter]</a>    <a href="chap3.html">[Next Chapter]</a>   </div>


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