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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 \(M_{24}\).

\(H^4(M_{24},\mathbb Z) = \mathbb Z_{12}\)

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 \(696729600\).

\(H_3(Weyl(E_8),\mathbb Z) = \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_{12}\)

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

\(H_4(Weyl(E_8),\mathbb Z) = \mathbb Z_2 \oplus \mathbb Z_2 \oplus Z_2 \oplus \mathbb Z_2\).

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);

Coxeter diagram for E8

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 \(M_{24}\).

\(H_6(Syl_2(M_{24}),\mathbb Z_2) = \mathbb Z_2^{143}\)

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

\(p(x)=\frac{1}{-x^3+3*x^2-3*x+1}\)

for the cohomology \(H^\ast(Syl_2(M_{12},\mathbb F_2)\). The coefficient of \(x^n\) in the expansion of \(p(x)\) is equal to the dimension of the vector space \(H^n(Syl_2(M_{12},\mathbb F_2)\). 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

\(p(x)=\frac{x^4-x^3+x^2-x+1}{x^6-x^5+x^4-2*x^3+x^2-x+1}\)

whose coefficient of \(x^n\) is equal to the dimension of the vector space \(H^n(M_{11},\mathbb F_2)\) for all \(n\) in the range \(0\le n\le 14\). The coefficient is not guaranteed correct for \(n\ge 15\).

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

\(H_4(N,\mathbb Z) = \mathbb (Z_3)^4 \oplus \mathbb Z^{84}\)

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

\(H_5(G,\mathbb Z) = \mathbb Z_2 \oplus \mathbb Z_2\)

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

\(H_6(SL_2({\cal O},\mathbb Z) = \mathbb Z_2\)

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

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

\(H_5(G,\mathbb Z) = \mathbb Z_3\)

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);;

Coxeter diagram for A7

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

6.6 Graphs of groups

The following example computes

\(H_5(G,\mathbb Z) = \mathbb Z_2\oplus Z_2\oplus Z_2 \oplus Z_2 \oplus Z_2\)

for \(G\) the graph of groups corresponding to the amalgamated product \(G=S_5*_{S_3}S_4\) 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);

graph of groups

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

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