  
  [1X9. Cohomology rings of p-groups (mainly p=2)[0X
  
  The functions on this page were written by Paul Smith. (They are included in
  HAP  but  they  are  also  independently  included  in  Paul Smiths HAPprime
  package.)
  
      | [10X Mod2CohomologyRingPresentation(G) [0X [10X Mod2CohomologyRingPresentation(G,n) [0X [10X Mod2CohomologyRingPresentation(A) [0X [10X Mod2CohomologyRingPresentation(R) [0X When applied to a finite 2-group G this function returns a presentation for the mod 2 cohomology ring H^*(G,Z_2). The Lyndon-Hochschild-Serre spectral sequence is used to prove that the presentation is correct. When the function is applied to a 2-group G and positive integer n the function first constructs n terms of a free Z_2G-resolution R, then constructs the finite-dimensional graded algebra A=H^(*le n)(G,Z_2), and finally uses A to approximate a presentation for H^*(G,Z_2). For "sufficiently large" the approximation will be a correct presentation for H^*(G,Z_2). Alternatively, the function can be applied directly to either the resolution R or graded algebra A. This function was written by Paul Smith. It uses the Singular commutative algebra package to handle the Lyndon-Hochschild-Serre spectral sequence. | 
      | [10XPoincareSeriesLHS(G) [0X Inputs a finite 2-group G and 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_2) for all k. This function was written by Paul Smith. It use the Singular system for commutative algebra.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | 
  
