<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.) <Section><Heading> &nbsp;</Heading> 
<ManSection> <Func Name="Mod2CohomologyRingPresentation" Arg="G"/> <Func Name="Mod2CohomologyRingPresentation" Arg="G,n"/> <Func Name="Mod2CohomologyRingPresentation" Arg="A"/> <Func Name="Mod2CohomologyRingPresentation" Arg="R"/> <Description> <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. <P/><B>Examples:</B> <URL><Link>../tutorial/chap8.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>2</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Var Name="PoincareSeriesLHS"/> <Description> <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. <P/><B>Examples:</B> 
</Description> </ManSection> </Section> </Chapter>