  
  [1X13 [33X[0;0YCohomology rings of [22Xp[122X[101X[1X-groups (mainly [22Xp=2)[122X[101X[1X[133X[101X
  
  [33X[0;0YThe functions on this page were written by [12XPaul Smith[112X. (They are included in
  HAP  but  they  are  also  independently  included  in  Paul Smiths HAPprime
  package.)[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XMod2CohomologyRingPresentation(G) [110X[133X
  [33X[0;0Y[10XMod2CohomologyRingPresentation(G,n) [110X[133X
  [33X[0;0Y[10XMod2CohomologyRingPresentation(A) [110X[133X
  [33X[0;0Y[10XMod2CohomologyRingPresentation(R) [110X[133X
  
  [33X[0;0YWhen  applied to a finite [22X2[122X-group [22XG[122X this function returns a presentation for
  the  mod  2 cohomology ring [22XH^*(G,Z_2)[122X. The Lyndon-Hochschild-Serre spectral
  sequence is used to prove that the presentation is correct.[133X
  
  [33X[0;0YWhen  the  function  is  applied  to  a [22X2[122X-group [22XG[122X and positive integer [22Xn[122X the
  function  first  constructs  [22Xn[122X  terms  of  a  free  [22XZ_2G[122X-resolution  [22XR[122X, then
  constructs  the  finite-dimensional  graded  algebra [22XA=H^(*le n)(G,Z_2)[122X, and
  finally   uses   [22XA[122X   to  approximate  a  presentation  for  [22XH^*(G,Z_2)[122X.  For
  "sufficiently  large"  the  approximation will be a correct presentation for
  [22XH^*(G,Z_2)[122X.[133X
  
  [33X[0;0YAlternatively, the function can be applied directly to either the resolution
  [22XR[122X or graded algebra [22XA[122X.[133X
  
  [33X[0;0YThis  function  was  written by [12XPaul Smith[112X. It uses the Singular commutative
  algebra package to handle the Lyndon-Hochschild-Serre spectral sequence.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XPoincareSeriesLHS(G) [110X[133X
  
  [33X[0;0YInputs   a   finite   [22X2[122X-group  [22XG[122X  and  returns  a  quotient  of  polynomials
  [22Xf(x)=P(x)/Q(x)[122X  whose coefficient of [22Xx^k[122X equals the rank of the vector space
  [22XH_k(G,Z_2)[122X for all [22Xk[122X.[133X
  
  [33X[0;0YThis  function  was  written  by  [12XPaul Smith[112X. It use the Singular system for
  commutative algebra.[133X
  
