GAP (HAP commands) - Chapter 6: Cohomology of groups

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6 Cohomology of groups
6.1 Finite groups
6.2 Nilpotent groups
6.3 Crystallographic groups
6.4 Arithmetic groups
6.5 Artin groups
6.6 Graphs of groups

### 6 Cohomology of groups

#### 6.1 Finite groups

The following example computes the fourth integral cohomomogy of the Mathieu group M_24.

H^4(M_24, Z) = Z_12

    gap> GroupCohomology(MathieuGroup(24),4);
    [ 4, 3 ]

The following example computes the third integral homology of the Weyl group W=Weyl(E_8), a group of order 696729600.

H_3(Weyl(E_8), Z) = Z_2 ⊕ Z_2 ⊕ Z_12

    p> L:=SimpleLieAlgebra("E",8,Rationals);;
    gap> W:=WeylGroup(RootSystem(L));;
    gap> Order(W);
    696729600
    gap> GroupHomology(W,3);
    [ 2, 2, 4, 3 ]

The preceding calculation could be achieved more quickly by noting that W=Weyl(E_8) is a Coxeter group, and by using the associated Coxeter polytope. The following example uses this approach to compute the fourth integral homology of W. It begins by displaying the Coxeter diagram of W, and then computes

H_4(Weyl(E_8), Z) = Z_2 ⊕ Z_2 ⊕ Z_2 ⊕ Z_2.

    gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3],[5,3]],[5,[6,3]],[6,[7,3]],[7,[8,3]]];;
    gap> CoxeterDiagramDisplay(D);

    gap> polytope:=CoxeterComplex_alt(D,5);;
    gap> R:=FreeGResolution(polytope,5);
    Resolution of length 5 in characteristic 0 for <matrix group with
    8 generators> .
    No contracting homotopy available.

    gap> C:=TensorWithIntegers(R);
    Chain complex of length 5 in characteristic 0 .

    gap> Homology(C,4);
    [ 2, 2, 2, 2 ]

The following example computes the sixth mod-2 homology of the Sylow 2-subgroup Syl_2(M_24) of the Mathieu group M_24.

H_6(Syl_2(M_24), Z_2) = Z_2^143

    gap> GroupHomology(SylowSubgroup(MathieuGroup(24),2),6,2);
    [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

The following example constructs the Poincare polynomial

p(x)=frac1-x^3+3*x^2-3*x+1

for the cohomology H^∗(Syl_2(M_12, F_2). The coefficient of x^n in the expansion of p(x) is equal to the dimension of the vector space H^n(Syl_2(M_12, F_2). The computation involves Singular's Groebner basis algorithms and the Lyndon-Hochschild-Serre spectral sequence.

    gap> G:=SylowSubgroup(MathieuGroup(12),2);;
    gap> PoincareSeriesLHS(G);
    (1)/(-x_1^3+3*x_1^2-3*x_1+1)

The following example constructs the polynomial

p(x)=fracx^4-x^3+x^2-x+1x^6-x^5+x^4-2*x^3+x^2-x+1

whose coefficient of x^n is equal to the dimension of the vector space H^n(M_11, F_2) for all n in the range 0le nle 14. The coefficient is not guaranteed correct for nge 15.

    gap> PoincareSeriesPrimePart(MathieuGroup(11),2,14);
    (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)

#### 6.2 Nilpotent groups

The following example computes

H_4(N, Z) = (Z_3)^4 ⊕ Z^84

for the free nilpotent group N of class 2 on four generators.

    gap> F:=FreeGroup(4);; N:=NilpotentQuotient(F,2);;
    gap> GroupHomology(N,4);
    [ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

#### 6.3 Crystallographic groups

The following example computes

H_5(G, Z) = Z_2 ⊕ Z_2

for the 3-dimensional crystallographic space group G with Hermann-Mauguin symbol "P62"

    gap> GroupHomology(SpaceGroupBBNWZ("P62"),5);
    [ 2, 2 ]

#### 6.4 Arithmetic groups

The following example computes

H_6(SL_2(cal O, Z) = Z_2

for cal O the ring of integers of the number field Q(sqrt-2).

    gap> C:=ContractibleGcomplex("SL(2,O-2)");;
    gap> R:=FreeGResolution(C,7);;
    gap> Homology(TensorWithIntegers(R),6);
    [ 2, 12 ]

#### 6.5 Artin groups

The following example computes

H_5(G, Z) = Z_3

for G the classical braid group on eight strings.

    gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,3]],[5,[6,3]],[6,[7,3]]];;
    gap> CoxeterDiagramDisplay(D);;

    gap> R:=ResolutionArtinGroup(D,6);;
    gap> C:=TensorWithIntegers(R);;
    gap> Homology(C,5);
    [ 3 ]

#### 6.6 Graphs of groups

The following example computes

H_5(G, Z) = Z_2⊕ Z_2⊕ Z_2 ⊕ Z_2 ⊕ Z_2

for G the graph of groups corresponding to the amalgamated product G=S_5*_S_3S_4 of the symmetric groups S_5 and S_4 over the canonical subgroup S_3.

    gap> S5:=SymmetricGroup(5);SetName(S5,"S5");
    gap> S4:=SymmetricGroup(4);SetName(S4,"S4");
    gap> A:=SymmetricGroup(3);SetName(A,"S3");
    gap> AS5:=GroupHomomorphismByFunction(A,S5,x->x);
    gap> AS4:=GroupHomomorphismByFunction(A,S4,x->x);
    gap> D:=[S5,S4,[AS5,AS4]];
    gap> GraphOfGroupsDisplay(D);

    gap> R:=ResolutionGraphOfGroups(D,6);;
    gap> Homology(TensorWithIntegers(R),5);
    [ 2, 2, 2, 2, 2 ]

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