<Chapter><Heading> Cohomology rings of <M>p</M>-groups (mainly <M>p=2)</M></Heading>

The functions on this page were written by <B>Paul Smith</B>. (They are included in  HAP but they are also independently included in Paul Smiths HAPprime package.)
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<Item>
<Index> Mod2CohomologyRingPresentation (HAPprime)</Index>
<C>Mod2CohomologyRingPresentation(G) </C>
<C>Mod2CohomologyRingPresentation(G,n) </C>
<C>Mod2CohomologyRingPresentation(A) </C>
<C>Mod2CohomologyRingPresentation(R) </C>

<P/>

 When applied to a finite <M>2</M>-group <M>G</M> this function returns a presentation for the mod 2 cohomology ring <M>H^*(G,Z_2)</M>. The Lyndon-Hochschild-Serre spectral sequence is used to prove that the presentation is correct.

<P/>
When the function is applied to a <M>2</M>-group <M>G</M> and positive integer <M>n</M> the function first constructs <M>n</M> terms of a free <M>Z_2G</M>-resolution <M>R</M>, then constructs the finite-dimensional graded algebra
<M>A=H^(*\le n)(G,Z_2)</M>, and finally uses <M>A</M> to approximate
 a presentation for <M>H^*(G,Z_2)</M>. For "sufficiently large" the approximation will be a correct presentation for <M>H^*(G,Z_2)</M>.

<P/> Alternatively, the function can be applied directly to either the resolution <M>R</M> or graded algebra <M>A</M>.

<P/>This function was written by <B>Paul Smith</B>. It uses the Singular commutative algebra package to handle the Lyndon-Hochschild-Serre spectral sequence.
</Item>
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<Row>
<Item>
<Index> PoincareSeriesLHS (HAPprime)</Index>
<C>PoincareSeriesLHS(G) </C>
<P/>

Inputs a finite <M>2</M>-group <M>G</M>
and returns a quotient of polynomials
<M>f(x)=P(x)/Q(x)</M> whose coefficient of <M>x^k</M>
equals the rank of the vector space <M>H_k(G,Z_2)</M>
for all <M>k</M>.
<P/>
This function was written by <B>Paul Smith</B>. It use the Singular system for commutative algebra.
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