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<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"><br /><span class="nocss"> </span><a href="chap2.html#X868E2A04832619C5">2.1-1 EquivariantChainMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X79EA11238403019D">2.1-2 FreeGResolution</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7CA87AA478007468">2.1-3 ResolutionBieberbachGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X81A5CEFC82A1897D">2.1-4 ResolutionCubicalCrystGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X789B3E7C7CBB3751">2.1-5 ResolutionFiniteGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7CBE6BDA7DB5AD7D">2.1-6 ResolutionNilpotentGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8574D76D7C891A04">2.1-7 ResolutionNormalSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X86934BE9858F7199">2.1-8 ResolutionPrimePowerGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7E4556B078B209CE">2.1-9 ResolutionSL2Z</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8518446086A3F7EA">2.1-10 ResolutionSmallGroup</a></span>
<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><a href="chap2.html#X85EC9D8E7A15A570">2.2 <span class="Heading"> Algebras <span class="SimpleMath">⟶</span> (Co)chain Complexes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D5DD19D7BA9D816">2.2-1 LeibnizComplex</a></span>
</div></div>
<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>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X788F3B5E7810E309">2.3-1 HomToIntegers</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X81FED0E9858E413A">2.3-2 HomToIntegralModule</a></span>
<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><a href="chap2.html#X8122D25786C83565">2.3-4 TensorWithIntegersModP</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X80B6849C835B7F19">2.4 <span class="Heading"> Cohomology rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X79C31EED8406A3E9">2.4-1 AreIsomorphicGradedAlgebras</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X83DC2F1A805BA7A3">2.4-2 HAPDerivation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7B93B7D082A50E61">2.4-3 HilbertPoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X803D9B5E7A26F749">2.4-4 HomologyOfDerivation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X855D2D747B6C54E1">2.4-5 IntegralCohomologyGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F5D00C97A46D686">2.4-6 LHSSpectralSequence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X828D20AC8735152B">2.4-7 LHSSpectralSequenceLastSheet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7DEFADD17CAA6308">2.4-8 ModPCohomologyGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X796632C585D47245">2.4-9 ModPCohomologyRing</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X831034A284F3906F">2.4-10 Mod2CohomologyRingPresentation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7BCF8D907D237A03">2.5 <span class="Heading"> Group Invariants</span></a>
</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>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X828B81D9829328F8">2.5-4 PoincareSeries</a></span>
<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 class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X84B5182E831D0928">2.6-2 Radical</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7929281B848A9FBE">2.6-3 RadicalSeries</a></span>
</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 covers the functions used in chapter 3 of the book <span class="URL"><a href="https://global.oup.com/academic/product/an-invitation-to-computational-homotopy-9780198832980">An Invitation to Computational Homotopy</a></span>.</p>
<p><a id="X7C0B125E7D5415B4" name="X7C0B125E7D5415B4"></a></p>
<h4>2.1 <span class="Heading"> Resolutions</span></h4>
<p><a id="X868E2A04832619C5" name="X868E2A04832619C5"></a></p>
<h5>2.1-1 EquivariantChainMap</h5>
<div class="func"><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 class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCohomologyRings.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeries.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutFunctorial.html">4</a></span> </p>
<p><a id="X79EA11238403019D" name="X79EA11238403019D"></a></p>
<h5>2.1-2 FreeGResolution</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGResolution</code>( <var class="Arg">P</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a non-free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">P_∗</span> and a positive integer <span class="SimpleMath">n</span>. It attempts to return <span class="SimpleMath">n</span> terms of a free <span class="SimpleMath">ZG</span>-resolution of <span class="SimpleMath">Z</span>. However, the stabilizer groups in the non-free resolution must be such that HAP can construct free resolutions with contracting homotopies for them.</p>
<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 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10
<p><a id="X7CA87AA478007468" namr>../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"><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">ZGjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<>
<pstrongbExamples> span=U>=./tutorialchap1span,<panclass"">ahref.tutorialhtml">2, >a href"../wwwSideLinksAbout.html">3, URL>< href./www/SideLinks/AboutaboutPerformancehtml">4 , URL><a href"./www/ideLinksAboutaboutPersistent./span> , span class="URL">./www/SideLinksAboutaboutPoincareSeries.html><a<s>,<span=URL<a ="..wwwSideLinksAbout/aboutDefinitions.html>7/aURL< href//SideLinksAboutaboutExtensions"8 , /span>
<p><a id="X80B6849C835B7F19" name="X80B6849C835B7F19"></a></p>
<<pan=ContSS >span="nocss>nbsp&;chap2htmlX81A5CEFC82A1897D.- ResolutionCubicalCrystGroup<s
<>aid"name=X79C31EED8406A3E9"<a<p>
<.4-1 AreIsomorphicGradedAlgebras</5>
<div="func"< ="func" ="10%"><><tdclass"func#227 AreIsomorphicGradedAlgebrascode(<var classArgA</var>,< class=Arg>B</var> )</>< ="">(nbsp&;)</d>/tr/ablediv
<p>Inputs two freely presented graded algebras><div
<p><strong/>
<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> )</</div</iv
<p>Inputs a polynomial ring <span class="SimpleMath">R= F/pan
<p><<span=""><br />span="nocss" </spanahrefchap2#"23HomToIntegralModule
</ivdivjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<h5>2.4-3 HilbertPoincareSeries</h5>
<divclassfunc>< class"" width"0%">tr classtdleftcodeclassfuncHilbertPoincareSeries</code < class="">E</var )/tdtd="tdright">(&bsp )/td></tr/ablediv
<p>Inputs a presentation <span class="SimpleMath">E= F[x_1,.<< ="">nbsp;<spanhref"hap2.#8DC2F1A805BA7A3"24- />/pan
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap8.html">1</a></span> </p>
<p< id=" name=X803D9B5E7A26F749>/a>/p>
<h5>2.4-4 HomologyOfDerivation<pan="ContSS">br/>span="nocss">&; /span><ahrefchap2#"24- LHSSpectralSequenceLastSheet
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><codespan="ContSS"<br />span="nocss" spanahref.#X796632C585D47245>.- ModPCohomologyRingspan
derivation classS":E →E/> on SimpleMath"=/I/> apolynomial < class=""R Fx_1,..,]/span> over fieldspan ="><span returns <span="impleMath">SJh</spanwhere< class"SimpleMath">S</span is a ring <spanclassSimpleMathJ<spanis list of foranidealin< class=""S<spansuch thatthere isan <spanclassSimpleMath>α SJ →ker/im/span This liftsto thering <spanclass="SimpleMath">h:S → d/>.This was written by Paul Smith. It uses the Singular commutative algebra package.</p>
<< class>xamples:strongp>
p< id="" name""><a><pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<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/>
< classContSSbr/< class"nocss"&bsp;</spana ="chap2.tmlX84B5182E831D0928"26-2 Radical</>/span>
<p><a id="X7F5D00C97A46D686" name="X7F5D00C97A46D686"></a></p>
<>. </>
=""="func" width0>tr ="tdleft">code=func27 /> var=ArgG/>,< =ArgN/> < =Argr/>)/>td=tdright(&;functionnbsp</d</tr/able/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>247</h5
class"unc>>&8227; LHSSpectralSequenceLastSheet</code(<var class=ArgG/>, <varclass="Arg>N >( &;)</></tr>/table>/>
<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"<p>Inputs a non-free <span class"SimpleMath"ZG<span-resolution < class="SimpleMath">P_∗/pananda integer< class"impleMath"n/span> It attemptsto return < class="SimpleMath>n
<p><strong class="button">Examples:</strong></p>
<p><a id="X7DEFADD17CAA6308" name="X7DEFADD17CAA6308"></a></p>
>. </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="‣ ResolutionBieberbachGroup( G, (nbsp )<td>/></table<div
divclass"">< class"unc width=10">tr ""< =func>&27 ModPCohomologyGenerators < class/>)td">&bspfunction&bsp;)
<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
lass:strong class href.www><>/> <p>
<p><a id="X796632C585D47245" name="X796632C585D47245"></a></p>
<h5>2.4-9 ModPCohomologyRing</h5>
<div class<p><a id"X789B3E7C7CBB3751 name="X789B3E7C7CBB3751>/a<p>
< class"func"width10%">tr>func82; /ode<var">/var>, level>)/> classtdright(nbspnbsp/></>/table/div
div=""><table="func width="0%><tr><td classtdleftcode class="">ModPCohomologyRing</code> <var<var var="Arg">n/var> )/td>< classtdright&;function;)</d></r></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></<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../tutorial/chap8.html">3</a></span> , <span class="URL"><a href=URL< href./tutorialchap6"1 spanclass">a ="../tutorial/chap7."><a></span>, < class""><ahref.//.html3</a</span> < class"URL"<="./tutorialchap10html>4 ,< "><a href="./tutorial/chap11.html"</a>/span, < classURL< href./www//."6/>s> spanclass="URL>< ="..www//bout/aboutPerformancehtml"7/>/>,span""a=.//SideLinksAbout/.html"8/a>>, ="">wwwSideLinks/.>/span< classURL=./About>0/>/> ,=" "//AboutaboutPoincareSeries"1/<> spanclass">ahrefSideLinksaboutCrossedMods12</>/span spanclass">a=.>.AboutaboutSimplicialGroupshtml>,span =URL>a =/.html"1/>/>,<span">.//About.">16, span class"">< ="..www/About.tml17</</>,span""ahref".wwwSideLinksA/."8<span spanclass">./wwwSideLinks//aboutTopologyhtml1</>/> ,< =URL"< =./SideLinks/aboutTwistedCoefficientshtml2/>/p>
<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"<p>Inputs groupspanclass"impleMath"G> (which beinfiniteand integer span="SimpleMath"> ≥1</span.It <span class="SimpleMath"k+/> terms a free < class=SimpleMathZG</>-resolution <span class"impleMath>Z.
<p< id=X8574D76D7C891A04name=X8574D76D7C891A04>/>/>
<p><a id>2.1- ResolutionNormalSeries</h5
<h5.-0Mod2CohomologyRingPresentation>
<div class="func"><table class="func" width="100%"><tr
<div class="func"><table class
< classfunc class"10"trclass">code ="func27Mod2CohomologyRingPresentation=Arg>/>)<t>td">&;functionnbsp)/> | /able
< div class= "func">< table class= "func" width= "100%">< tr>< td class= "tdleft">< code class= "func">‣ Mo d2CohomologyRingPresentation</code>( <var class="Arg">R</var> )</td><td class="tdright">( function > a <span=SimpleMath</pan < =""><> an< class"k /pan. < class""+/> of free< class=S"<>resolution field=> of class"">/>.</>
<p>When applied toafinite <span="SimpleMath"></>-group<spanclass=SimpleMath>G/> this returnsapresentation mod-<panclass"SimpleMath"2</>cohomology <span=SimpleMath>H∗,F</span Lyndon-Hochschild-Serrespectralsequenceisusedtothatthepresentationis . When functionisappliedto spanclassSimpleMath></span- Gandpositiveinteger<span class"impleMath"><span the functionfirst constructsspan class"">n1span terms a freespan class=SimpleMath>FG/>-resolutionspanclass""><span> then thefinite-dimensionalgradedalgebra < class"">=^( ≤n)G, F)</>,and finallyuses<pan="SimpleMath> to approximate a presentation SimpleMath>^G </>. sufficientlylarge" spanclass"">n the approximation will a correct presentation for H^(G F</ thefunctioncan applied resolutions =">R or graded algebra A. This function was written by Paul Smith. It uses the Singular commutative algebra package to handle the Lyndon-Hochschild-Serre spectral sequence.
<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>
paid"="<></p>
<h4>2.5 <span class="Heading"> Group Invariants</span></h4>
<p><a id="X7D1658EF810022E5" name="X7D1658EF810022E5"><
<h5>2pThis was by Kholodna<p>
<
<div class="func"><table ="func>table=func =10%>tr>(>R/>, < classvar> ="">&;function<></></></>
< a ZG-resolutionofspan="Z/> and index subgroupH ≤ G/span>. Itreturns a "><>
<p>If <><trong=buttonExamples/ span=URLa =".wwwSideLinksAboutaboutArithmetichtml"<>span =URLa=.//SideLinks/aboutArtinGroups></a<span span=""><a ="../www/SideLinks/About/boutTwistedCoefficients.html">3</a></span> </p>
<>strong="button">:</strong>=htmla> span=URL.html//> />
<p><a id="X7F0A19E97980FD57" name="<>> class="0%>tr>td =t"><ode=func#27 </code var=Arg/>, < class><var/><tdclass"tdright"(nbspnbsptd>/>/iv
<div class="func"><table class="func" width="
<p>Inputs <h4. <span classHeading Resolutions<span="SimpleMath>⟶/span> (Co)chain Complexes
<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="URL"><a href="../tutorial/chap13.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLinks.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutParallel.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRosenbergerMonster.html">5</a></span> , <spandiv="func"table=funcwidth0%><> class">func">&827; HomToIntegers((nbspfunctionnbsp;
<p><a id
<h5>2.5-3 PrimePartDerivedFunctor</h5>
<div
<p>Inputs a group <<div class="func"><table"10"<><td="">< class"func"#227; HomToIntegralModule>< =Arg/ar<class">A/var> )>(td<tr/able>
<p><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X828B81D9829328F8" name="X828B81D9829328F8"></a></p>
<h5>2.5-4 PoincareSeries</h5>
<div class="func"><table class="func" width="100%"><<div class="func"><tableclass width="10%"<trtd="tdleft">< class"func"> TensorWithIntegers/ode <var="Arg">R</> )/td< class"">( </d><tr<table<>
<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 <p>Inputs a free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMat> characteristicclassSimpleMath">0/span>and returnsthechaincomplex>R⊗ <span/>
< class"unc"> class"func"width"00%"<trtd="tdleft"><code class"func>̻PoincareSeriesArg"> class=Arg> | >(nbsp )</d><tr/table>>
></panItreturns quotientof span=""fx)P()/Q(x</>whose="xk"H_kG, ) forall< ="SimpleMathk/span inthe <span="">1 k ≤n<span>.Thesecond variablecanbeomitted inwhichcasethefunction to choose a `reasonable valuefor< =""n/pan For2groups function <strong="">oincareSeriesLHSG)<strongcanbeusedtoproduce spanclass"SimpleMath">(x/>that correctinalldegrees thegroups classSimpleMath>G</span thefunction also(tleastspanclass"impleMath"</> of mod-spanclassSimpleMathp</panresolution <spanclass"SimpleMath"R<span for <spanclassG>.Alternatively,thefirst input can bea list<span class"SimpleMath"L</span of. this the of<span class"SimpleMath"xk/span> < class=SimpleMath>(x)/span>is to the <span class="SimpleMath">(k+1)</span>st term in the list.</p>
<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="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
<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=<>When to <pan=""></span>groupspan="SimpleMath>G/span>thisfunctionreturns presentationthe mod- =""<>cohomology class">∗G )span Lyndon-Hochschild-Serre that .When spanclassSimpleMath<- integer =<>the constructs ="n<> s =""FG/>resolution spanclassSR/, the finite-dimensionalgraded spanclassSimpleMath>=^ n(,F, uses< =SimpleMath>/>toapproximatea span="impleMathH,/> sufficiently ="the presentation SimpleMathH∗,<> function resolution=SimpleMath<spangradedclass></> It handle /java.lang.StringIndexOutOfBoundsException: Index 1338 out of bounds for length 1338
<p><a id="X7EFE814686C4EEF5" name="X7EFE814686C4EEF5"></a></p>
<h5>2.5-6 RankHomologyPGroup</h5>
<div="">< class"width"10%>tr class"tdleft"< classfunc#27 RankHomologyPGroupcode < classArg">G, Pvar">< class="tdright&<>t>/<>
<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
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> </p>
p<="X86CDD4B77CBE3087"nameX86CDD4B77CBE3087/>/p>
<h4>2.6 <span class="Heading"> <span class="SimpleMath">F_p</span>-modules</span></h4>
<p>a id"name"">/
<h5h5>2.5- GroupHomologyh5
<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=
<p>Inputs an <span class="SimpleMath">F_pG</span>-module and returns <div class="func"><table class="func" width="100%"><tr><td class="tdleft <>< =Arg><varvarA""><varv class<> )</td class>&;function;)/d<tr></div
<p><strong class="button">Examples:</strong></p>
<p><a idX7929281B848A9FBE 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>
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