  
  [1X6 [33X[0;0YCohomology of groups[133X[101X
  
  
  [1X6.1 [33X[0;0YFinite groups[133X[101X
  
  [33X[0;0YThe following example computes the fourth integral cohomomogy of the Mathieu
  group [22XM_24[122X.[133X
  
  [33X[0;0Y[22XH^4(M_24, Z) = Z_12[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGroupCohomology(MathieuGroup(24),4);[127X[104X
    [4X[28X[ 4, 3 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe following example computes the third integral homology of the Weyl group
  [22XW=Weyl(E_8)[122X, a group of order [22X696729600[122X.[133X
  
  [33X[0;0Y[22XH_3(Weyl(E_8), Z) = Z_2 ⊕ Z_2 ⊕ Z_12[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28Xp> L:=SimpleLieAlgebra("E",8,Rationals);;[128X[104X
    [4X[25Xgap>[125X [27XW:=WeylGroup(RootSystem(L));;[127X[104X
    [4X[25Xgap>[125X [27XOrder(W);[127X[104X
    [4X[28X696729600[128X[104X
    [4X[25Xgap>[125X [27XGroupHomology(W,3);[127X[104X
    [4X[28X[ 2, 2, 4, 3 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  preceding  calculation  could  be  achieved more quickly by noting that
  [22XW=Weyl(E_8)[122X  is  a  Coxeter  group,  and  by  using  the  associated Coxeter
  polytope.  The  following  example  uses this approach to compute the fourth
  integral  homology  of  [22XW[122X. It begins by displaying the Coxeter diagram of [22XW[122X,
  and then computes[133X
  
  [33X[0;0Y[22XH_4(Weyl(E_8), Z) = Z_2 ⊕ Z_2 ⊕ Z_2 ⊕ Z_2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:=[[1,[2,3]],[2,[3,3]],[3,[4,3],[5,3]],[5,[6,3]],[6,[7,3]],[7,[8,3]]];;[127X[104X
    [4X[25Xgap>[125X [27XCoxeterDiagramDisplay(D);[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpolytope:=CoxeterComplex_alt(D,5);;[127X[104X
    [4X[25Xgap>[125X [27XR:=FreeGResolution(polytope,5);[127X[104X
    [4X[28XResolution of length 5 in characteristic 0 for <matrix group with [128X[104X
    [4X[28X8 generators> . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XC:=TensorWithIntegers(R);[127X[104X
    [4X[28XChain complex of length 5 in characteristic 0 . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XHomology(C,4);[127X[104X
    [4X[28X[ 2, 2, 2, 2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  example  computes  the  sixth  mod-[22X2[122X  homology  of the Sylow
  [22X2[122X-subgroup [22XSyl_2(M_24)[122X of the Mathieu group [22XM_24[122X.[133X
  
  [33X[0;0Y[22XH_6(Syl_2(M_24), Z_2) = Z_2^143[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGroupHomology(SylowSubgroup(MathieuGroup(24),2),6,2);[127X[104X
    [4X[28X[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, [128X[104X
    [4X[28X  2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, [128X[104X
    [4X[28X  2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, [128X[104X
    [4X[28X  2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, [128X[104X
    [4X[28X  2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, [128X[104X
    [4X[28X  2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, [128X[104X
    [4X[28X  2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe following example constructs the Poincare polynomial[133X
  
  [33X[0;0Y[22Xp(x)=frac1-x^3+3*x^2-3*x+1[122X[133X
  
  [33X[0;0Yfor  the  cohomology  [22XH^∗(Syl_2(M_12,  F_2)[122X.  The  coefficient of [22Xx^n[122X in the
  expansion   of   [22Xp(x)[122X  is  equal  to  the  dimension  of  the  vector  space
  [22XH^n(Syl_2(M_12,  F_2)[122X.  The  computation  involves [12XSingular[112X's Groebner basis
  algorithms and the Lyndon-Hochschild-Serre spectral sequence.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SylowSubgroup(MathieuGroup(12),2);;[127X[104X
    [4X[25Xgap>[125X [27XPoincareSeriesLHS(G);[127X[104X
    [4X[28X(1)/(-x_1^3+3*x_1^2-3*x_1+1)[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe following example constructs the polynomial[133X
  
  [33X[0;0Y[22Xp(x)=fracx^4-x^3+x^2-x+1x^6-x^5+x^4-2*x^3+x^2-x+1[122X[133X
  
  [33X[0;0Ywhose  coefficient  of  [22Xx^n[122X  is  equal  to the dimension of the vector space
  [22XH^n(M_11,  F_2)[122X  for  all  [22Xn[122X in the range [22X0le nle 14[122X. The coefficient is not
  guaranteed correct for [22Xnge 15[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPoincareSeriesPrimePart(MathieuGroup(11),2,14);[127X[104X
    [4X[28X(x_1^4-x_1^3+x_1^2-x_1+1)/(x_1^6-x_1^5+x_1^4-2*x_1^3+x_1^2-x_1+1)[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.2 [33X[0;0YNilpotent groups[133X[101X
  
  [33X[0;0YThe following example computes[133X
  
  [33X[0;0Y[22XH_4(N, Z) = (Z_3)^4 ⊕ Z^84[122X[133X
  
  [33X[0;0Yfor the free nilpotent group [22XN[122X of class [22X2[122X on four generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup(4);; N:=NilpotentQuotient(F,2);;[127X[104X
    [4X[25Xgap>[125X [27XGroupHomology(N,4);[127X[104X
    [4X[28X[ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.3 [33X[0;0YCrystallographic groups[133X[101X
  
  [33X[0;0YThe following example computes[133X
  
  [33X[0;0Y[22XH_5(G, Z) = Z_2 ⊕ Z_2[122X[133X
  
  [33X[0;0Yfor  the  [22X3[122X-dimensional  crystallographic space group [22XG[122X with Hermann-Mauguin
  symbol "P62"[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGroupHomology(SpaceGroupBBNWZ("P62"),5);[127X[104X
    [4X[28X[ 2, 2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.4 [33X[0;0YArithmetic groups[133X[101X
  
  [33X[0;0YThe following example computes[133X
  
  [33X[0;0Y[22XH_6(SL_2(cal O, Z) = Z_2[122X[133X
  
  [33X[0;0Yfor [22Xcal O[122X the ring of integers of the number field [22XQ(sqrt-2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XC:=ContractibleGcomplex("SL(2,O-2)");;[127X[104X
    [4X[25Xgap>[125X [27XR:=FreeGResolution(C,7);;[127X[104X
    [4X[25Xgap>[125X [27XHomology(TensorWithIntegers(R),6);[127X[104X
    [4X[28X[ 2, 12 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.5 [33X[0;0YArtin groups[133X[101X
  
  [33X[0;0YThe following example computes[133X
  
  [33X[0;0Y[22XH_5(G, Z) = Z_3[122X[133X
  
  [33X[0;0Yfor [22XG[122X the classical braid group on eight strings.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,3]],[5,[6,3]],[6,[7,3]]];;[127X[104X
    [4X[25Xgap>[125X [27XCoxeterDiagramDisplay(D);;[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionArtinGroup(D,6);;[127X[104X
    [4X[25Xgap>[125X [27XC:=TensorWithIntegers(R);;[127X[104X
    [4X[25Xgap>[125X [27XHomology(C,5);[127X[104X
    [4X[28X[ 3 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.6 [33X[0;0YGraphs of groups[133X[101X
  
  [33X[0;0YThe following example computes[133X
  
  [33X[0;0Y[22XH_5(G, Z) = Z_2⊕ Z_2⊕ Z_2 ⊕ Z_2 ⊕ Z_2[122X[133X
  
  [33X[0;0Yfor  [22XG[122X  the  graph  of  groups  corresponding  to  the  amalgamated  product
  [22XG=S_5*_S_3S_4[122X  of  the  symmetric  groups  [22XS_5[122X  and  [22XS_4[122X  over the canonical
  subgroup [22XS_3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS5:=SymmetricGroup(5);SetName(S5,"S5");[127X[104X
    [4X[25Xgap>[125X [27XS4:=SymmetricGroup(4);SetName(S4,"S4");[127X[104X
    [4X[25Xgap>[125X [27XA:=SymmetricGroup(3);SetName(A,"S3");[127X[104X
    [4X[25Xgap>[125X [27XAS5:=GroupHomomorphismByFunction(A,S5,x->x);[127X[104X
    [4X[25Xgap>[125X [27XAS4:=GroupHomomorphismByFunction(A,S4,x->x);[127X[104X
    [4X[25Xgap>[125X [27XD:=[S5,S4,[AS5,AS4]];[127X[104X
    [4X[25Xgap>[125X [27XGraphOfGroupsDisplay(D);[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionGraphOfGroups(D,6);;[127X[104X
    [4X[25Xgap>[125X [27XHomology(TensorWithIntegers(R),5);[127X[104X
    [4X[28X[ 2, 2, 2, 2, 2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
