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<div class="ChapSects"><a href="chap7.html#X850CDAFE801E2B2A">7. <span class="Heading"> Poincare series</span></a>
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<h3>7. <span class="Heading"> Poincare series</span></h3>
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<td class="tdleft"><code class="code">EfficientNormalSubgroups(G)</code> <br /> <code class="code">EfficientNormalSubgroups(G,k)</code></p>
<p>Inputs a prime-power group G and, optionally, a positive integer k. The default is k=4. The function returns a list of normal subgroups N in G such that the Poincare series for G equals the Poincare series for the direct product (N x (G/N)) up to degree k.</td>
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<td class="tdleft"><code class="code">ExpansionOfRationalFunction(f,n)</code></p>
<p>Inputs a positive integer n and a rational function f(x)=p(x)/q(x) where the degree of the polynomial p(x) is less than that of q(x). It returns a list [a_0 , a_1 , a_2 , a_3 , ... ,a_n] of the first n+1 coefficients of the infinite expansion</p>
<p>f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ... .</td>
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<td class="tdleft"><code class="code"> PoincareSeries(G,n) </code> <code class="code"> PoincareSeries(R,n) </code> <br /> <code class="code"> PoincareSeries(L,n) </code> <br /> <code class="code"> PoincareSeries(G) </code></p>
<p>Inputs a finite p-group G and a positive integer n. It returns a quotient of polynomials f(x)=P(x)/Q(x) whose coefficient of x^k equals the rank of the vector space H_k(G,Z_p) for all k in the range k=1 to k=n. (The second input variable can be omitted, in which case the function tries to choose a "reasonable" value for n.)</p>
<p>In place of the group G the function can also input (at least n terms of) a minimal mod p resolution R for G.</p>
<p>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.</td>
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<td class="tdleft"><code class="code">PoincareSeriesPrimePart(G,p,n) </code></p>
<p>Inputs a finite group G, a prime p, and a positive integer n. It returns a quotient of polynomials f(x)=P(x)/Q(x) whose coefficient of x^k equals the rank of the vector space H_k(G,Z_p) for all k in the range k=1 to k=n.</p>
<p>The efficiency of this function needs to be improved.</td>
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<td class="tdleft"><code class="code"> Prank(G) </code></p>
<p>Inputs a p-group G and returns the rank of the largest elementary abelian subgroup.</td>
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