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<Chapter><Heading> Poincare series</Heading>

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<Table Align="|l|" >

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<Row>

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<Item>

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<C>EfficientNormalSubgroups(G)</C>

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<C>EfficientNormalSubgroups(G,k)</C>

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<P/>

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Inputs a prime-power group <M>G</M> and, optionally, a positive 

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integer <M>k</M>. The default is <M>k=4</M>. The function returns a 

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list of normal subgroups <M>N</M> in <M>G</M> such that the Poincare 

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series for <M>G</M> equals 

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the Poincare series for the direct product <M>(N \times (G/N))</M> up to degree <M>k</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> ExpansionOfRationalFunction</Index>

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<C>ExpansionOfRationalFunction(f,n)</C>

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<P/>

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Inputs a positive integer <M>n</M> and a rational function 

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<M>f(x)=p(x)/q(x)</M> where the degree of the polynomial <M>p(x)</M>

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is less than that of <M>q(x)</M>. It returns a list 

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<M>[a_0 , a_1 , a_2 , a_3 , \ldots ,a_n]</M> of the first <M>n+1</M>

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coefficients of the infinite expansion

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<P/>

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<M>f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots </M>  .

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> PoincareSeries</Index>

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<C>PoincareSeries(G,n) </C>

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<C> PoincareSeries(R,n) </C>

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<C> PoincareSeries(L,n) </C>

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<C> PoincareSeries(G) </C>

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<P/>

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Inputs a finite <M>p</M>-group <M>G</M> and a positive integer <M>n</M>. 

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It returns a quotient of polynomials <M>f(x)=P(x)/Q(x)</M> 

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whose coefficient of <M>x^k</M> equals the rank of the vector space 

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<M>H_k(G,Z_p)</M> for all <M>k</M> in the range <M>k=1</M> to <M>k=n</M>.  

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(The second input variable can be omitted, in which case the 

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function tries to choose a "reasonable" value for <M>n</M>. For <M>2</M>-groups the function PoincareSeriesLHS(G) can be used to produce an <M>f(x)</M> that

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is correct in all degrees.)

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<P/>

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In place of the group <M>G</M> the function can also input 

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(at least <M>n</M> terms of) a minimal mod <M>p</M> resolution 

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<M>R</M> for <M>G</M>.

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<P/>

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Alternatively, the first input variable can be a list <M>L</M>

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of integers. In this case the coefficient of <M>x^k</M> in <M>f(x)</M>

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is equal to the <M>(k+1)</M>st term in the list.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> PoincareSeriesPrimePart</Index>

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<C>PoincareSeriesPrimePart(G,p,n) </C>

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<P/>

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Inputs a finite group <M>G</M>, a prime <M>p</M>, and a positive integer 

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<M>n</M>. It returns a quotient of polynomials 

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<M>f(x)=P(x)/Q(x)</M> whose coefficient of <M>x^k</M> 

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equals the rank of the vector space <M>H_k(G,Z_p)</M> 

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for all <M>k</M> in the range <M>k=1</M> to <M>k=n</M>.

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<P/>

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The efficiency of this function needs to be improved.

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</Item>

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</Row>

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<Row>

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<Item>

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<C>PoincareSeriesLHS(G) </C>

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<P/>

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Inputs a finite <M>2</M>-group <M>G</M>

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and returns a quotient of polynomials

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<M>f(x)=P(x)/Q(x)</M> whose coefficient of <M>x^k</M>

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equals the rank of the vector space <M>H_k(G,Z_2)</M>

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for all <M>k</M>.

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<P/>

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This function was written by <B>Paul Smith</B>. It use the Singular system for commutative algebra. 

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> Prank</Index>

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<C>Prank(G) </C>

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<P/>

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Inputs a <M>p</M>-group <M>G</M> and returns the rank of the 

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largest elementary abelian subgroup. 

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</Item>

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</Row>

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</Table>

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</Chapter>

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