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<title>GAP (HAP) - Chapter 15: FpG-modules</title>
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<div class="ChapSects"><a href="chap15.html#X81A2A3C97C09685E">15. <span class="Heading"> FpG-modules</span></a>
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<h3>15. <span class="Heading"> FpG-modules</span></h3>
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<td class="tdleft"><code class="code"> CompositionSeriesOfFpGModules(M) </code></p>
<p>Inputs an FpG-module M and returns a list of FpG-modules that constitute a composition series for M.</td>
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<td class="tdleft"><code class="code"> DirectSumOfFpGModules(M,N) </code> <br /> <code class="code"> DirectSumOfFpGModules([ M[1], M[2], ..., M[k] ])) </code></p>
<p>Inputs two FpG-modules M and N with common group and characteristic. It returns the direct sum of M and N as an FpG-Module.</p>
<p>Alternatively, the function can input a list of FpG-modules with common group G. It returns the direct sum of the list.</td>
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<td class="tdleft"><code class="code"> FpGModule(A,P) </code> <br /> <code class="code"> FpGModule(A,G,p) </code></p>
<p>Inputs a p-group P and a matrix A whose rows have length a multiple of the order of G. It returns the "canonical" FpG-module generated by the rows of A.</p>
<p>A small non-prime-power group G can also be input, provided the characteristic p is entered as a third input variable.</td>
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<td class="tdleft"><code class="code"> FpGModuleDualBasis(M) </code></p>
<p>Inputs an FpG-module M. It returns a record R with two components:</p>
<ul>
<li><p>R.freeModule is the free module FG of rank one.</p>
</li>
<li><p>R.basis is a list representing an F-basis for the module Hom_FG(M,FG). Each term in the list is a matrix A whose rows are vectors in FG such that M!.generators[i] --> A[i] extends to a module homomorphism M --> FG.</p>
</li>
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<td class="tdleft"><code class="code"> FpGModuleHomomorphism(M,N,A) </code> <br /> <code class="code"> FpGModuleHomomorphismNC(M,N,A) </code></p>
<p>Inputs FpG-modules M and N over a common p-group G. Also inputs a list A of vectors in the vector space spanned by N!.matrix. It tests that the function</p>
<p>M!.generators[i] --> A[i]</p>
<p>extends to a homomorphism of FpG-modules and, if the test is passed, returns the corresponding FpG-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 FpG-module homomorphism.</td>
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<td class="tdleft"><code class="code"> DesuspensionFpGModule(M,n)</code> <br /> <code class="code"> DesuspensionFpGModule(R,n) </code></p>
<p>Inputs a positive integer n and and FpG-module M. It returns an FpG-module D^nM which is mathematically related to M via an exact sequence 0 --> D^nM --> R_n --> ... --> R_0 --> M --> 0 where R_* is a free resolution. (If G=Group(M) is of prime-power order then the resolution is minimal.)</p>
<p>Alternatively, the function can input a positive integer n and at least n terms of a free resolution R of M.</td>
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<td class="tdleft"><code class="code"> RadicalOfFpGModule(M) </code></p>
<p>Inputs an FpG-module M with G a p-group, and returns the Radical of M as an FpG-module. (Ig G is not a p-group then a submodule of the radical is returned.</td>
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<td class="tdleft"><code class="code"> RadicalSeriesOfFpGModule(M) </code></p>
<p>Inputs an FpG-module M and returns a list of FpG-modules that constitute the radical series for M.</td>
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<td class="tdleft"><code class="code"> GeneratorsOfFpGModule(M) </code></p>
<p>Inputs an FpG-module M and returns a matrix whose rows correspond to a minimal generating set for M.</td>
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<td class="tdleft"><code class="code"> ImageOfFpGModuleHomomorphism(f) </code></p>
<p>Inputs an FpG-module homomorphism f:M --> N and returns its image f(M) as an FpG-module.</td>
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<td class="tdleft"><code class="code"> GroupAlgebraAsFpGModule(G) </code></p>
<p>Inputs a p-group G and returns its mod p group algebra as an FpG-module.</td>
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<td class="tdleft"><code class="code"> IntersectionOfFpGModules(M,N) </code></p>
<p>Inputs two FpG-modules M, N arising as submodules in a common free module (FG)^n where G is a finite group and F the field of p-elements. It returns the FpG-module arising as the intersection of M and N.</td>
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<td class="tdleft"><code class="code"> IsFpGModuleHomomorphismData(M,N,A) </code></p>
<p>Inputs FpG-modules M and N over a common p-group G. Also inputs a list A of vectors in the vector space spanned by N!.matrix. It returns true if the function</p>
<p>M!.generators[i] --> A[i]</p>
<p>extends to a homomorphism of FpG-modules. Otherwise it returns false.</td>
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<td class="tdleft"><code class="code"> MaximalSubmoduleOfFpGModule(M) </code></p>
<p>Inputs an FpG-module M and returns one maximal FpG-submodule of M.</td>
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<td class="tdleft"><code class="code"> MaximalSubmodulesOfFpGModule(M) </code></p>
<p>Inputs an FpG-module M and returns the list of maximal FpG-submodules of M.</td>
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<td class="tdleft"><code class="code"> MultipleOfFpGModule(w,M) </code></p>
<p>Inputs an FpG-module M and a list w:=[g_1 , ..., g_t] of elements in the group G=M!.group. The list w can be thought of as representing the element w=g_1 + ... + g_t in the group algebra FG, and the function returns a semi-echelon matrix B which is a basis for the vector subspace wM .</td>
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<td class="tdleft"><code class="code"> ProjectedFpGModule(M,k) </code></p>
<p>Inputs an FpG-module M of ambient dimension n|G|, and an integer k between 1 and n. The module M is a submodule of the free module (FG)^n . Let M_k denote the intersection of M with the last k summands of (FG)^n . The function returns the image of the projection of M_k onto the k-th summand of (FG)^n . This image is returned an FpG-module with ambient dimension |G|.</td>
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<td class="tdleft"><code class="code"> RandomHomomorphismOfFpGModules(M,N) </code></p>
<p>Inputs two FpG-modules M and N over a common group G. It returns a random matrix A whose rows are vectors in N such that the function</p>
<p>M!.generators[i] --> A[i]</p>
<p>extends to a homomorphism M --> N of FpG-modules. (There is a problem with this function at present.)</td>
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<td class="tdleft"><code class="code"> Rank(f) </code></p>
<p>Inputs an FpG-module homomorphism f:M --> N and returns the dimension of the image of f as a vector space over the field F of p elements.</td>
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<td class="tdleft"><code class="code"> SumOfFpGModules(M,N) </code></p>
<p>Inputs two FpG-modules M, N arising as submodules in a common free module (FG)^n where G is a finite group and F the field of p-elements. It returns the FpG-Module arising as the sum of M and N.</td>
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<td class="tdleft"><code class="code"> SumOp(f,g) </code></p>
<p>Inputs two FpG-module homomorphisms f,g:M --> N with common sorce and common target. It returns the sum f+g:M --> N . (This operation is also available using "+".</td>
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<td class="tdleft"><code class="code"> VectorsToFpGModuleWords(M,L) </code></p>
<p>Inputs an FpG-module M and a list L=[v_1,... ,v_k] of vectors in M. It returns a list L'= [x_1,...,x_k] . Each x_j=[[W_1,G_1],...,[W_t,G_t]] is a list of integer pairs corresponding to an expression of v_j as a word</p>
<p>v_j = g_1*w_1 + g_2*w_1 + ... + g_t*w_t</p>
<p>where</p>
<p>g_i=Elements(M!.group)[G_i]</p>
<p>w_i=GeneratorsOfFpGModule(M)[W_i] .</td>
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