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<div class="ChapSects"><a href="chap21_mj.html#X81A2A3C97C09685E">21 <span class="Heading"> <span class="SimpleMath">\(FpG\)</span>-modules</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21_mj.html#X7CFDEEC07F15CF82">21.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X78A33AD27F99EF01">21.1-1 CompositionSeriesOfFpGModules</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X7BECAE4182051B42">21.1-2 DirectSumOfFpGModules</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X7BADE250862DDD5F">21.1-3 FpGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X834924DF873A86B3">21.1-4 FpGModuleDualBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X851DB60D862610A5">21.1-5 FpGModuleHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X8380187081CE1237">21.1-6 DesuspensionFpGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X87A9EA448236280E">21.1-7 RadicalOfFpGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X87B150217D5AB909">21.1-8 RadicalSeriesOfFpGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X7CEC4AFB8181D098">21.1-9 GeneratorsOfFpGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X7AC4426279F2C004">21.1-10 ImageOfFpGModuleHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X85758F95832207D2">21.1-11 GroupAlgebraAsFpGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X83D91CE68582A650">21.1-12 IntersectionOfFpGModules</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X7F1A1FA280D8C5D0">21.1-13 IsFpGModuleHomomorphismData</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X78F8BC20784828A7">21.1-14 MaximalSubmoduleOfFpGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X85C8FD707F10C22F">21.1-15 MaximalSubmodulesOfFpGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X7C94F977801C1B97">21.1-16 MultipleOfFpGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X804F7A1B79BEC521">21.1-17 ProjectedFpGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X822272997C4A8352">21.1-18 RandomHomomorphismOfFpGModules</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X827146F37E2AA841">21.1-19 Rank</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X7A6C598E82D5A7F8">21.1-20 SumOfFpGModules</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X86CAE52783D8E343">21.1-21 SumOp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap21_mj.html#X8290448D87105F0B">21.1-22 VectorsToFpGModuleWords</a></span>
</div></div>
</div>
<h3>21 <span class="Heading"> <span class="SimpleMath">\(FpG\)</span>-modules</span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>21.1 <span class="Heading"> </span></h4>
<p><a id="X78A33AD27F99EF01" name="X78A33AD27F99EF01"></a></p>
<h5>21.1-1 CompositionSeriesOfFpGModules</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CompositionSeriesOfFpGModules</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module <span class="SimpleMath">\(M\)</span> and returns a list of <span class="SimpleMath">\(FpG\)</span>-modules that constitute a composition series for <span class="SimpleMath">\(M\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7BECAE4182051B42" name="X7BECAE4182051B42"></a></p>
<h5>21.1-2 DirectSumOfFpGModules</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DirectSumOfFpGModules</code>( <var class="Arg">M</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">&#an style='color: green'>8227; DirectSumOfFpGModules</code>( [<var class="Arg">M</var>[, <var class="Arg">1</var>], <var class="Arg">M</var>[, <var class="Arg">2</var>], <var class="Arg">...</var>, <var class="Arg">M</var>[, <var class="Arg">k</var>]] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two <span class="SimpleMath">\(FpG\)</span>-modules <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(N\)</span> with common group and characteristic. It returns the direct sum of <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(N\)</span> as an <span class="SimpleMath">\(FpG\)</span>-Module.</p>
<p>Alternatively, the function can input a list of <span class="SimpleMath">\(FpG\)</span>-modules with common group <span class="SimpleMath">\(G\)</span>. It returns the direct sum of the list.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7BADE250862DDD5F" name="X7BADE250862DDD5F"></a></p>
<h5>21.1-3 FpGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FpGModule</code>( <var class="Arg">A</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">&#an style='color: green'>8227; FpGModule</code>( <var class="Arg">A</var>, <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(P\)</span> and a matrix <span class="SimpleMath">\(A\)</span> whose rows have length a multiple of the order of <span class="SimpleMath">\(G\)</span>. It returns the <q>canonical</q> <span class="SimpleMath">\(FpG\)</span>-module generated by the rows of <span class="SimpleMath">\(A\)</span>.</p>
<p>A small non-prime-power group <span class="SimpleMath">\(G\)</span> can also be input, provided the characteristic <span class="SimpleMath">\(p\)</span> is entered as a third input variable.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTorAndExt.html">4</a></span> </p>
<p><a id="X834924DF873A86B3" name="X834924DF873A86B3"></a></p>
<h5>21.1-4 FpGModuleDualBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FpGModuleDualBasis</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module <span class="SimpleMath">\(M\)</span>. It returns a record <span class="SimpleMath">\(R\)</span> with two components:</p>
<ul>
<li><p><span class="SimpleMath">\(R.freeModule\)</span> is the free module <span class="SimpleMath">\(FG\)</span> of rank one.</p>
</li>
<li><p><span class="SimpleMath">\(R.basis\)</span> is a list representing an <span class="SimpleMath">\(F\)</span>-basis for the module <span class="SimpleMath">\(Hom_{FG}(M,FG)\)</span>. Each term in the list is a matrix <span class="SimpleMath">\(A\)</span> whose rows are vectors in <span class="SimpleMath">\(FG\)</span> such that <span class="SimpleMath">\(M!.generators[i] \longrightarrow A[i]\)</span> extends to a module homomorphism <span class="SimpleMath">\(M \longrightarrow FG\)</span>.</p>
</li>
</ul>
<p><strong class="button">Examples:</strong></p>
<p><a id="X851DB60D862610A5" name="X851DB60D862610A5"></a></p>
<h5>21.1-5 FpGModuleHomomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FpGModuleHomomorphism</code>( <var class="Arg">M</var>, <var class="Arg">N</var>, <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">&#an style='color: green'>8227; FpGModuleHomomorphismNC</code>( <var class="Arg">M</var>, <var class="Arg">N</var>, <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs <span class="SimpleMath">\(FpG\)</span>-modules <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(N\)</span> over a common <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span>. Also inputs a list <span class="SimpleMath">\(A\)</span> of vectors in the vector space spanned by <span class="SimpleMath">\(N!.matrix\)</span>. It tests that the function</p>
<p><span class="SimpleMath">\( M!.generators[i] \longrightarrow A[i]\)</span></p>
<p>extends to a homomorphism of <span class="SimpleMath">\(FpG\)</span>-modules and, if the test is passed, returns the corresponding <span class="SimpleMath">\(FpG\)</span>-module homomorphism. If the test is failed it returns fail.</p>
<p>The "NC" version of the function assumes that the input defines a homomorphism and simply returns the <span class="SimpleMath">\(FpG\)</span>-module homomorphism.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X8380187081CE1237" name="X8380187081CE1237"></a></p>
<h5>21.1-6 DesuspensionFpGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DesuspensionFpGModule</code>( <var class="Arg">M</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">&#an style='color: green'>8227; DesuspensionFpGModule</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a positive integer <span class="SimpleMath">\(n\)</span> and and FpG-module <span class="SimpleMath">\(M\)</span>. It returns an FpG-module <span class="SimpleMath">\(D^nM\)</span> which is mathematically related to <span class="SimpleMath">\(M\)</span> via an exact sequence <span class="SimpleMath">\( 0 \longrightarrow D^nM \longrightarrow R_n \longrightarrow \ldots \longrightarrow R_0 \longrightarrow M \longrightarrow 0\)</span> where <span class="SimpleMath">\(R_\ast\)</span> is a free resolution. (If <span class="SimpleMath">\(G=Group(M)\)</span> is of prime-power order then the resolution is minimal.)</p>
<p>Alternatively, the function can input a positive integer <span class="SimpleMath">\(n\)</span> and at least <span class="SimpleMath">\(n\)</span> terms of a free resolution <span class="SimpleMath">\(R\)</span> of <span class="SimpleMath">\(M\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X87A9EA448236280E" name="X87A9EA448236280E"></a></p>
<h5>21.1-7 RadicalOfFpGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RadicalOfFpGModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module <span class="SimpleMath">\(M\)</span> with <span class="SimpleMath">\(G\)</span> a <span class="SimpleMath">\(p\)</span>-group, and returns the Radical of <span class="SimpleMath">\(M\)</span> as an <span class="SimpleMath">\(FpG\)</span>-module. (Ig <span class="SimpleMath">\(G\)</span> is not a <span class="SimpleMath">\(p\)</span>-group then a submodule of the radical is returned.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X87B150217D5AB909" name="X87B150217D5AB909"></a></p>
<h5>21.1-8 RadicalSeriesOfFpGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RadicalSeriesOfFpGModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module <span class="SimpleMath">\(M\)</span> and returns a list of <span class="SimpleMath">\(FpG\)</span>-modules that constitute the radical series for <span class="SimpleMath">\(M\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7CEC4AFB8181D098" name="X7CEC4AFB8181D098"></a></p>
<h5>21.1-9 GeneratorsOfFpGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GeneratorsOfFpGModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module <span class="SimpleMath">\(M\)</span> and returns a matrix whose rows correspond to a minimal generating set for <span class="SimpleMath">\(M\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7AC4426279F2C004" name="X7AC4426279F2C004"></a></p>
<h5>21.1-10 ImageOfFpGModuleHomomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ImageOfFpGModuleHomomorphism</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module homomorphism <span class="SimpleMath">\(f:M \longrightarrow N\)</span> and returns its image <span class="SimpleMath">\(f(M)\)</span> as an <span class="SimpleMath">\(FpG\)</span>-module.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X85758F95832207D2" name="X85758F95832207D2"></a></p>
<h5>21.1-11 GroupAlgebraAsFpGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GroupAlgebraAsFpGModule</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span> and returns its mod <span class="SimpleMath">\(p\)</span> group algebra as an <span class="SimpleMath">\(FpG\)</span>-module.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X83D91CE68582A650" name="X83D91CE68582A650"></a></p>
<h5>21.1-12 IntersectionOfFpGModules</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IntersectionOfFpGModules</code>( <var class="Arg">M</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two <span class="SimpleMath">\(FpG\)</span>-modules <span class="SimpleMath">\(M, N\)</span> arising as submodules in a common free module <span class="SimpleMath">\((FG)^n\)</span> where <span class="SimpleMath">\(G\)</span> is a finite group and <span class="SimpleMath">\(F\)</span> the field of <span class="SimpleMath">\(p\)</span>-elements. It returns the <span class="SimpleMath">\(FpG\)</span>-module arising as the intersection of <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(N\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7F1A1FA280D8C5D0" name="X7F1A1FA280D8C5D0"></a></p>
<h5>21.1-13 IsFpGModuleHomomorphismData</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsFpGModuleHomomorphismData</code>( <var class="Arg">M</var>, <var class="Arg">N</var>, <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs <span class="SimpleMath">\(FpG\)</span>-modules <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(N\)</span> over a common <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span>. Also inputs a list <span class="SimpleMath">\(A\)</span> of vectors in the vector space spanned by <span class="SimpleMath">\(N!.matrix\)</span>. It returns true if the function</p>
<p><span class="SimpleMath">\( M!.generators[i] \longrightarrow A[i]\)</span></p>
<p>extends to a homomorphism of <span class="SimpleMath">\(FpG\)</span>-modules. Otherwise it returns false.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X78F8BC20784828A7" name="X78F8BC20784828A7"></a></p>
<h5>21.1-14 MaximalSubmoduleOfFpGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MaximalSubmoduleOfFpGModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module <span class="SimpleMath">\(M\)</span> and returns one maximal <span class="SimpleMath">\(FpG\)</span>-submodule of <span class="SimpleMath">\(M\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X85C8FD707F10C22F" name="X85C8FD707F10C22F"></a></p>
<h5>21.1-15 MaximalSubmodulesOfFpGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MaximalSubmodulesOfFpGModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module <span class="SimpleMath">\(M\)</span> and returns the list of maximal <span class="SimpleMath">\(FpG\)</span>-submodules of <span class="SimpleMath">\(M\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7C94F977801C1B97" name="X7C94F977801C1B97"></a></p>
<h5>21.1-16 MultipleOfFpGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MultipleOfFpGModule</code>( <var class="Arg">w</var>, <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module <span class="SimpleMath">\(M\)</span> and a list <span class="SimpleMath">\(w:=[g_1 , ..., g_t]\)</span> of elements in the group <span class="SimpleMath">\(G=M!.group\)</span>. The list <span class="SimpleMath">\(w\)</span> can be thought of as representing the element <span class="SimpleMath">\(w=g_1 + \ldots + g_t\)</span> in the group algebra <span class="SimpleMath">\(FG\)</span>, and the function returns a semi-echelon matrix <span class="SimpleMath">\(B\)</span> which is a basis for the vector subspace <span class="SimpleMath">\(wM\)</span> .</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X804F7A1B79BEC521" name="X804F7A1B79BEC521"></a></p>
<h5>21.1-17 ProjectedFpGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectedFpGModule</code>( <var class="Arg">M</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module <span class="SimpleMath">\(M\)</span> of ambient dimension <span class="SimpleMath">\(n|G|\)</span>, and an integer <span class="SimpleMath">\(k\)</span> between <span class="SimpleMath">\(1\)</span> and <span class="SimpleMath">\(n\)</span>. The module <span class="SimpleMath">\(M\)</span> is a submodule of the free module <span class="SimpleMath">\((FG)^n\)</span> . Let <span class="SimpleMath">\(M_k\)</span> denote the intersection of <span class="SimpleMath">\(M\)</span> with the last <span class="SimpleMath">\(k\)</span> summands of <span class="SimpleMath">\((FG)^n\)</span> . The function returns the image of the projection of <span class="SimpleMath">\(M_k\)</span> onto the <span class="SimpleMath">\(k\)</span>-th summand of <span class="SimpleMath">\((FG)^n\)</span> . This image is returned an <span class="SimpleMath">\(FpG\)</span>-module with ambient dimension <span class="SimpleMath">\(|G|\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X822272997C4A8352" name="X822272997C4A8352"></a></p>
<h5>21.1-18 RandomHomomorphismOfFpGModules</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RandomHomomorphismOfFpGModules</code>( <var class="Arg">M</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two <span class="SimpleMath">\(FpG\)</span>-modules <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(N\)</span> over a common group <span class="SimpleMath">\(G\)</span>. It returns a random matrix <span class="SimpleMath">\(A\)</span> whose rows are vectors in <span class="SimpleMath">\(N\)</span> such that the function</p>
<p><span class="SimpleMath">\( M!.generators[i] \longrightarrow A[i]\)</span></p>
<p>extends to a homomorphism <span class="SimpleMath">\(M \longrightarrow N\)</span> of <span class="SimpleMath">\(FpG\)</span>-modules. (There is a problem with this function at present.)</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X827146F37E2AA841" name="X827146F37E2AA841"></a></p>
<h5>21.1-19 Rank</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Rank</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module homomorphism <span class="SimpleMath">\(f:M \longrightarrow N\)</span> and returns the dimension of the image of <span class="SimpleMath">\(f\)</span> as a vector space over the field <span class="SimpleMath">\(F\)</span> of <span class="SimpleMath">\(p\)</span> elements.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPeriodic.html">3</a></span> </p>
<p><a id="X7A6C598E82D5A7F8" name="X7A6C598E82D5A7F8"></a></p>
<h5>21.1-20 SumOfFpGModules</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SumOfFpGModules</code>( <var class="Arg">M</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two <span class="SimpleMath">\(FpG\)</span>-modules <span class="SimpleMath">\(M, N\)</span> arising as submodules in a common free module <span class="SimpleMath">\((FG)^n\)</span> where <span class="SimpleMath">\(G\)</span> is a finite group and <span class="SimpleMath">\(F\)</span> the field of <span class="SimpleMath">\(p\)</span>-elements. It returns the <span class="SimpleMath">\(FpG\)</span>-Module arising as the sum of <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(N\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X86CAE52783D8E343" name="X86CAE52783D8E343"></a></p>
<h5>21.1-21 SumOp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SumOp</code>( <var class="Arg">f</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two <span class="SimpleMath">\(FpG\)</span>-module homomorphisms <span class="SimpleMath">\(f,g:M \longrightarrow N\)</span> with common sorce and common target. It returns the sum <span class="SimpleMath">\(f+g:M \longrightarrow N\)</span> . (This operation is also available using "+".</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X8290448D87105F0B" name="X8290448D87105F0B"></a></p>
<h5>21.1-22 VectorsToFpGModuleWords</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; VectorsToFpGModuleWords</code>( <var class="Arg">M</var>, <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(FpG\)</span>-module <span class="SimpleMath">\(M\)</span> and a list <span class="SimpleMath">\(L=[v_1,\ldots ,v_k]\)</span> of vectors in <span class="SimpleMath">\(M\)</span>. It returns a list <span class="SimpleMath">\(L'= [x_1,...,x_k]\)</span> . Each <span class="SimpleMath">\(x_j=[[W_1,G_1],...,[W_t,G_t]]\)</span> is a list of integer pairs corresponding to an expression of <span class="SimpleMath">\(v_j\)</span> as a word</p>
<p><span class="SimpleMath">\( v_j = g_1*w_1 + g_2*w_1 + ... + g_t*w_t \)</span></p>
<p>where</p>
<p><span class="SimpleMath">\(g_i=Elements(M!.group)[G_i]\)</span></p>
<p><span class="SimpleMath">\(w_i=GeneratorsOfFpGModule(M)[W_i]\)</span> .</p>
<p><strong class="button">Examples:</strong></p>
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