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11 Resolutions
11.1 Resolutions for small finite groups
11.2 Resolutions for very small finite groups
11.3 Resolutions for finite groups acting on orbit polytopes
11.4 Minimal resolutions for finite p-groups over F_p
11.5 Resolutions for abelian groups
11.6 Resolutions for nilpotent groups
11.7 Resolutions for groups with subnormal series
11.8 Resolutions for groups with normal series
11.9 Resolutions for polycyclic (almost) crystallographic groups
11.10 Resolutions for Bieberbach groups
11.11 Resolutions for arbitrary crystallographic groups
11.12 Resolutions for crystallographic groups admitting cubical fundamental domain
11.13 Resolutions for Coxeter groups
11.14 Resolutions for Artin groups
11.15 Resolutions for G=SL_2( Z[1/m])
11.16 Resolutions for selected groups G=SL_2( mathcal O( Q(sqrtd) )
11.17 Resolutions for selected groups G=PSL_2( mathcal O( Q(sqrtd) )
11.18 Resolutions for a few higher-dimensional arithmetic groups
11.19 Resolutions for finite-index subgroups
11.20 Simplifying resolutions
11.21 Resolutions for graphs of groups and for groups with aspherical presentations
11.22 Resolutions for FG-modules

### 11 Resolutions

There is a range of functions in HAP that input a group G, integer n, and attempt to return the first n terms of a free ZG-resolution R_∗ of the trivial module Z. In some cases an explicit contracting homotopy is provided on the resolution. The function `Size(R)` returns a list whose kth term is the sum of the lengths of the boundaries of the generators in degree k.

#### 11.1 Resolutions for small finite groups

The following uses discrete Morse theory to construct a resolution.

    gap> G:=SymmetricGroup(6);; n:=6;;
    gap> R:=ResolutionFiniteGroup(G,n);
    Resolution of length 6 in characteristic 0 for Group([ (1,2), (1,2,3,4,5,6)
     ]) .

    gap> Size(R);
    [ 10, 58, 186, 452, 906, 1436 ]

#### 11.2 Resolutions for very small finite groups

The following uses linear algebra over Z to construct a resolution.

    gap> Q:=QuaternionGroup(128);;
    gap> R:=ResolutionSmallGroup(Q,20);
    Resolution of length 20 in characteristic 0 for <pc group of size 128 with
    2 generators> .
    No contracting homotopy available.

    gap> Size(R);
    [ 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128 ]

The suspicion that this resolution R_∗ is periodic of period 4 can be verified by constructing the chain complex C_∗=R_∗⊗_ Z ZG and verifying that boundary matrices repeat with period 4.

A second example of a periodic resolution, for the Dihedral group D_2k+1=⟨ x, y | x^2= xy^kx^-1y^-k-1⟩ of order 2k+2 in the case k=1, is constructed and verified for periodicity in the next example.

    gap> R:=ResolutionSmallGroup(D,15);;
    gap> Size(R);
    [ 4, 7, 8, 6, 4, 8, 8, 6, 4, 8, 8, 6, 4, 8, 8 ]
    gap> C:=TensorWithIntegersOverSubgroup(R,Group(One(D)));;
    gap> n:=4;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);
    true
    gap> n:=5;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);
    true
    gap> n:=6;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);
    true
    gap> n:=7;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);
    true
    gap> n:=8;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);
    true

This periodic resolution for D_3 can be found in a paper by R. Swan [Swa60]. The resolution was proved for arbitrary D_2k+1 by Irina Kholodna [Kho01] (Corollary 5.5) and is the cellular chain complex of the universal cover of a CW-complex X with two cells in dimensions 1, 2 mod 4 and one cell in dimensions 0,3 mod 4. The 2-skelecton is the 2-complex for the given presentation of D_2k+1 and an attaching map for the 3-cell is represented as follows.

A slightly different periodic resolution for D_2k+1 has been obtain more recently by FEA Johnson [Joh16]. Johnson's resolution has two free generators in each degree. Interestingly, running the following code for many values of k >1 seems to produce a periodic resolution with two free generators in each degree for most values of k.

    gap> k:=20;;rels:=[x^2,x*y^k*x^-1*y^(-1-k)];;D:=F/rels;;
    gap> R:=ResolutionSmallGroup(D,7);;
    gap> List([0..7],R!.dimension);
    [ 1, 2, 2, 2, 2, 2, 2, 2 ]

#### 11.3 Resolutions for finite groups acting on orbit polytopes

The following uses Polymake convex hull computations and homological perturbation theory to construct a resolution.

    gap> G:=SignedPermutationGroup(5);;
    gap> StructureDescription(G);
    "C2 x ((C2 x C2 x C2 x C2) : S5)"

    gap> v:=[1,2,3,4,5];;  #The resolution depends on the choice of vector.
    gap> P:=PolytopalComplex(G,[1,2,3,4,5]);
    Non-free resolution in characteristic 0 for <matrix group of size 3840 with
    9 generators> .
    No contracting homotopy available.

    gap> R:=FreeGResolution(P,6);
    Resolution of length 5 in characteristic 0 for <matrix group of size
    3840 with 9 generators> .
    No contracting homotopy available.
    gap> Size(R);
    [ 10, 60, 214, 694, 6247, 273600 ]

The convex polytope P_G(v)= Convex~Hull{g⋅ v | g∈ G} used in the resolution depends on the choice of vector v∈ R^n. Two such polytopes for the alternating group G=A_4 acting on R^4 can be visualized as follows.

    gap> G:=AlternatingGroup(4);;
    gap> OrbitPolytope(G,[1,2,3,4],["VISUAL"]);
    gap> OrbitPolytope(G,[1,1,3,4],["VISUAL"]);

    gap> P1:=PolytopalComplex(G,[1,2,3,4]);;
    gap> P2:=PolytopalComplex(G,[1,1,3,4]);;
    gap> R1:=FreeGResolution(P1,20);;
    gap> R2:=FreeGResolution(P2,20);;
    gap> Size(R1);
    [ 6, 11, 32, 24, 36, 60, 65, 102, 116, 168, 172, 248, 323, 628, 650, 1093,
      1107, 2456, 2344, 6115 ]
    gap> Size(R2);
    [ 4, 11, 20, 24, 36, 60, 65, 102, 116, 168, 172, 248, 323, 628, 650, 1093,
      1107, 2456, 2344, 6115 ]

#### 11.4 Minimal resolutions for finite p-groups over F_p

The following uses linear algebra to construct a minimal free F_pG-resolution of the trivial module F.

    gap> P:=SylowSubgroup(MathieuGroup(12),2);;
    gap> R:=ResolutionPrimePowerGroup(P,20);
    Resolution of length 20 in characteristic 2 for Group(
    [ (2,8,4,12)(3,11,7,9), (2,3)(4,7)(6,10)(9,11), (3,7)(6,10)(8,11)(9,12),
      (1,10)(3,7)(5,6)(8,12), (2,4)(3,7)(8,12)(9,11), (1,5)(6,10)(8,12)(9,11)
     ]) .

    gap> Size(R);
    [ 6, 62, 282, 740, 1810, 3518, 6440, 10600, 17040, 24162, 34774, 49874,
      62416, 81780, 106406, 145368, 172282, 208926, 262938, 320558 ]

The resolution has the minimum number of generators possible in each degree and can be used to guess a formula for the Poincare series

P(x) = Σ_k≥ 0 dim_ F_pH^k(G, F_p)x^k.

The guess is certainly correct for the coefficients of x^k for k≤ 20 and can be used to guess the dimension of say H^2000(G, F_p).

Most likely dim_ F_2H^2000(G, F_2) = 2001000.

    gap> P:=PoincareSeries(R,20);
    (1)/(-x_1^3+3*x_1^2-3*x_1+1)

    gap> ExpansionOfRationalFunction(P,2000)[2000];
    2001000

#### 11.5 Resolutions for abelian groups

The following uses the formula for the tensor product of chain complexes to construct a resolution.

    gap> A:=AbelianPcpGroup([2,4,8,0,0]);;
    gap> StructureDescription(A);
    "Z x Z x C8 x C4 x C2"

    gap> R:=ResolutionAbelianGroup(A,10);
    Resolution of length 10 in characteristic 0 for Pcp-group with orders
    [ 2, 4, 8, 0, 0 ] .

    gap> Size(R);
    [ 14, 90, 296, 680, 1256, 2024, 2984, 4136, 5480, 7016 ]

#### 11.6 Resolutions for nilpotent groups

The following uses the NQ package to express the free nilpotent group of class 3 on three generators as a Pcp group G, and then uses homological perturbation on the lower central series to construct a resolution. The resolution is used to exhibit 2-torsion in H_4(G, Z).

    gap> F:=FreeGroup(3);;
    gap> G:=Image(NqEpimorphismNilpotentQuotient(F,3));;
    gap> R:=ResolutionNilpotentGroup(G,5);
    Resolution of length 5 in characteristic 0 for Pcp-group with orders
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] .

    gap> Size(R);
    [ 28, 377, 2377, 9369, 25850 ]

    gap> Homology(TensorWithIntegers(R),4);
    [ 2, 2, 2, 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, 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, 0, 0, 0 ]

The following uses homological perturbation on the lower central series to construct a resolution for the Sylow 2-subgroup P=Syl_2(M_12) of the Mathieu simple group M_12.

    gap> G:=MathieuGroup(12);;
    gap> P:=SylowSubgroup(G,2);;
    gap> StructureDescription(P);
    "((C4 x C4) : C2) : C2"

    gap> R:=ResolutionNilpotentGroup(P,9);
    Resolution of length 9 in characteristic
    0 for <permutation group with 279 generators> .

    gap> Size(R);
    [ 12, 80, 310, 939, 2556, 6768, 19302, 61786, 237068 ]

#### 11.7 Resolutions for groups with subnormal series

The following uses homological perturbation on a subnormal series to construct a resolution for the Sylow 2-subgroup P=Syl_2(M_12) of the Mathieu simple group M_12.

    gap> P:=SylowSubgroup(MathieuGroup(12),2);;
    gap> sn:=ElementaryAbelianSeries(P);;
    gap> R:=ResolutionSubnormalSeries(sn,9);
    Resolution of length 9 in characteristic
    0 for <permutation group with 64 generators> .

    gap> Size(R);
    [ 12, 78, 288, 812, 1950, 4256, 8837, 18230, 39120 ]

#### 11.8 Resolutions for groups with normal series

The following uses homological perturbation on a normal series to construct a resolution for the Sylow 2-subgroup P=Syl_2(M_12) of the Mathieu simple group M_12.

    gap> P:=SylowSubgroup(MathieuGroup(12),2);;
    gap> P1:=EfficientNormalSubgroups(P)[1];;
    gap> P2:=Intersection(DerivedSubgroup(P),P1);;
    gap> P3:=Group(One(P));;
    gap> R:=ResolutionNormalSeries([P,P1,P2,P3],9);
    Resolution of length 9 in characteristic
    0 for <permutation group with 64 generators> .

    gap> Size(R);
    [ 10, 60, 200, 532, 1238, 2804, 6338, 15528, 40649 ]

#### 11.9 Resolutions for polycyclic (almost) crystallographic groups

The following uses the Polycyclic package and homological perturbation to construct a resolution for the crystallographic group `G:=SpaceGroup(3,165)`.

    gap> G:=SpaceGroup(3,165);;
    gap> G:=Image(IsomorphismPcpGroup(G));;
    gap> R:=ResolutionAlmostCrystalGroup(G,20);
    Resolution of length 20 in characteristic 0 for Pcp-group with orders
    [ 3, 2, 0, 0, 0 ] .

    gap> Size(R);
    [ 10, 49, 117, 195, 273, 351, 429, 507, 585, 663, 741, 819, 897, 975, 1053,
      1131, 1209, 1287, 1365, 1443 ]

The following constructs a resolution for an almost crystallographic Pcp group G. The final commands establish that G is not isomorphic to a crystallographic group.

    gap> G:=AlmostCrystallographicPcpGroup( 4, 50, [ 1, -4, 1, 2 ] );;
    gap> R:=ResolutionAlmostCrystalGroup(G,20);
    Resolution of length 20 in characteristic 0 for Pcp-group with orders
    [ 4, 0, 0, 0, 0 ] .

    gap> Size(R);
    [ 10, 53, 137, 207, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223,
      223, 223, 223, 223, 223 ]

    gap> T:=Kernel(NaturalHomomorphismOnHolonomyGroup(G));;
    gap> IsAbelian(T);
    false

#### 11.10 Resolutions for Bieberbach groups

The following constructs a resolution for the Bieberbach group `G=SpaceGroup(3,165)` by using convex hull algorithms to construct a Dirichlet domain for its free action on Euclidean space R^3. By construction the resolution is trivial in degrees ≥ 3.

    gap> G:=SpaceGroup(3,165);;
    gap> R:=ResolutionBieberbachGroup(G);
    Resolution of length 4 in characteristic
    0 for SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 ) .
    No contracting homotopy available.

    gap> Size(R);
    [ 10, 18, 8, 0 ]

The fundamental domain constructed for the above resolution can be visualized using the following commands.

    gap> F:=FundamentalDomainBieberbachGroup(G);
    <polymake object>
    gap> Display(F);

A different fundamental domain and resolution for G can be obtained by changing the choice of vector v∈ R^3 in the definition of the Dirichlet domain

D(v) = {x∈ R^3 | ||x-v|| ≤ ||x-g.v|| for~all~ g∈ G}.

    gap> R:=ResolutionBieberbachGroup(G,[1/2,1/2,1/2]);
    Resolution of length 4 in characteristic
    0 for SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 ) .
    No contracting homotopy available.

    gap> Size(R);
    [ 28, 42, 16, 0 ]

    gap> F:=FundamentalDomainBieberbachGroup(G);
    <polymake object>
    gap> Display(F);

A higher dimensional example is handled in the next session. A list of the 62 7-dimensional Hantze-Wendt Bieberbach groups is loaded and a resolution is computed for the first group in the list.

    gap> file:=HapFile("HW-7dim.txt");;
    gap> Read(file);
    gap> G:=HWO7Gr[1];
    <matrix group with 7 generators>

    gap> R:=ResolutionBieberbachGroup(G);
    Resolution of length 8 in characteristic 0 for <matrix group with
    7 generators> .
    No contracting homotopy available.

    gap> Size(R);
    [ 284, 1512, 3780, 4480, 2520, 840, 84, 0 ]

The homological perturbation techniques needed to extend this method to crystallographic groups acting non-freely on R^n has not yet been implemenyed. This is on the TO-DO list.

#### 11.11 Resolutions for arbitrary crystallographic groups

An implementation of the above method for Bieberbach groups is also available for arbitrary crystallographic groups. The following example constructs a resolution for the group `G:=SpaceGroupIT(3,227)`.

    gap> G:=SpaceGroupIT(3,227);;
    gap> R:=ResolutionSpaceGroup(G,11);
    Resolution of length 11 in characteristic 0 for <matrix group with
    8 generators> .
    No contracting homotopy available.

    gap> Size(R);
    [ 38, 246, 456, 644, 980, 1427, 2141, 2957, 3993, 4911, 6179 ]

#### 11.12 Resolutions for crystallographic groups admitting cubical fundamental domain

The following uses subdivision techniques to construct a resolution for the Bieberbach group `G:=SpaceGroup(4,122)`. The resolution is endowed with a contracting homotopy.

    gap> G:=SpaceGroup(4,122);;
    gap> R:=ResolutionCubicalCrystGroup(G,20);
    Resolution of length 20 in characteristic 0 for <matrix group with
    6 generators> .

    gap> Size(R);
    [ 8, 24, 24, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

Subdivision and homological perturbation are used to construct the following resolution (with contracting homotopy) for a crystallographic group with non-free action.

    gap> G:=SpaceGroup(4,1100);;
    gap> R:=ResolutionCubicalCrystGroup(G,20);
    Resolution of length 20 in characteristic 0 for <matrix group with
    8 generators> .

    gap> Size(R);
    [ 40, 215, 522, 738, 962, 1198, 1466, 1734, 2034, 2334, 2666, 2998, 3362,
      3726, 4122, 4518, 4946, 5374, 5834, 6294 ]

#### 11.13 Resolutions for Coxeter groups

The following session constructs the Coxeter diagram for the Coxeter group B=B_7 of order 645120. A resolution for G is then computed.

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

    gap> R:=ResolutionCoxeterGroup(D,5);
    Resolution of length 5 in characteristic
    0 for <permutation group of size 645120 with 7 generators> .
    No contracting homotopy available.

    gap> Size(R);
    [ 14, 112, 492, 1604, 5048 ]

The routine extension of this method to infinite Coxeter groups is on the TO-DO list.

#### 11.14 Resolutions for Artin groups

The following session constructs a resolution for the infinite Artin group G associated to the Coxeter group B_7. Exactness of the resolution depends on the solution to the K(π,1) Conjecture for Artin groups of spherical type.

    gap> R:=ResolutionArtinGroup(D,8);
    Resolution of length 8 in characteristic 0 for <fp group on the generators
    [ f1, f2, f3, f4, f5, f6, f7 ]> .
    No contracting homotopy available.

    gap> Size(R);
    [ 14, 98, 310, 610, 918, 1326, 2186, 0 ]

#### 11.15 Resolutions for G=SL_2( Z[1/m])

The following uses homological perturbation to construct a resolution for G=SL_2( Z[1/6]).

    gap> R:=ResolutionSL2Z(6,10);
    Resolution of length 10 in characteristic 0 for SL(2,Z[1/6]) .

    gap> Size(R);
    [ 44, 679, 6910, 21304, 24362, 48506, 43846, 90928, 86039, 196210 ]

#### 11.16 Resolutions for selected groups G=SL_2( mathcal O( Q(sqrtd) )

The following uses finite "Voronoi complexes" and homological perturbation to construct a resolution for G=SL_2(mathcal O( Q(sqrt-5)). The finite complexes were contributed independently by A. Rahm, M. Dutour-Scikiric and S. Schoenenbeck and are stored in the folder `~pkg/Hap1.v/lib/Perturbations/Gcomplexes`.

    gap> R:=ResolutionSL2QuadraticIntegers(-5,10);
    Resolution of length 10 in characteristic 0 for matrix group .
    No contracting homotopy available.

    gap> Size(R);
    [ 22, 114, 120, 200, 146, 156, 136, 254, 168, 170 ]

#### 11.17 Resolutions for selected groups G=PSL_2( mathcal O( Q(sqrtd) )

The following uses finite "Voronoi complexes" and homological perturbation to construct a resolution for G=PSL_2(mathcal O( Q(sqrt-11)). The finite complexes were contributed independently by A. Rahm, M. Dutour-Scikiric and S. Schoenenbeck and are stored in the folder `~pkg/Hap1.v/lib/Perturbations/Gcomplexes`.

    gap> R:=ResolutionPSL2QuadraticIntegers(-11,10);
    Resolution of length 10 in characteristic 0 for PSL(2,O-11) .
    No contracting homotopy available.

    gap> Size(R);
    [ 12, 59, 89, 107, 125, 230, 208, 270, 326, 515 ]

#### 11.18 Resolutions for a few higher-dimensional arithmetic groups

The following uses finite "Voronoi complexes" and homological perturbation to construct a resolution for G=PSL_4( Z). The finite complexes were contributed by M. Dutour-Scikiric and are stored in the folder `~pkg/Hap1.v/lib/Perturbations/Gcomplexes`.

    gap>  V:=ContractibleGcomplex("PSL(4,Z)_d");
    Non-free resolution in characteristic 0 for matrix group .
    No contracting homotopy available.

    gap> R:=FreeGResolution(V,5);
    Resolution of length 5 in characteristic 0 for matrix group .
    No contracting homotopy available.

    gap> Size(R);
    [ 18, 210, 1444, 26813 ]

#### 11.19 Resolutions for finite-index subgroups

The next commands first construct the congruence subgroup Γ_0(I) of index 144 in SL_2(cal O Q(sqrt-2)) for the ideal I in cal O Q(sqrt-2) generated by 4+5sqrt-2. The commands then compute a resolution for the congruence subgroup G=Γ_0(I) ≤ SL_2(cal O Q(sqrt-2))

    gap> Q:=QuadraticNumberField(-2);;
    gap> OQ:=RingOfIntegers(Q);;
    gap> I:=QuadraticIdeal(OQ,4+5*Sqrt(-2));;
    gap> G:=HAP_CongruenceSubgroupGamma0(I);
    <[group of 2x2 matrices in characteristic 0>
    gap>
    gap> IndexInSL2O(G);
    144
    gap> R:=ResolutionSL2QuadraticIntegers(-2,4,true);;
    gap> S:=ResolutionFiniteSubgroup(R,G);
    Resolution of length 4 in characteristic 0 for <matrix group with
    290 generators> .

    gap> Size(S);
    [ 1152, 8496, 30960, 59616 ]

#### 11.20 Simplifying resolutions

The next commands construct a resolution R_∗ for the symmetric group S_5 and convert it to a resolution S_∗ for the finite index subgroup A_4 < S_5. An heuristic algorithm is applied to S_∗ in the hope of obtaining a smaller resolution T_∗ for the alternating group A_4.

    gap> R:=ResolutionFiniteGroup(SymmetricGroup(5),5);;
    gap> S:=ResolutionFiniteSubgroup(R,AlternatingGroup(4));
    Resolution of length 5 in characteristic 0 for Alt( [ 1 .. 4 ] ) .

    gap> Size(S);
    [ 80, 380, 1000, 2040, 3400 ]
    gap> T:=SimplifiedComplex(S);
    Resolution of length 5 in characteristic 0 for Alt( [ 1 .. 4 ] ) .

    gap> Size(T);
    [ 4, 34, 22, 19, 196 ]

#### 11.21 Resolutions for graphs of groups and for groups with aspherical presentations

The following example constructs a resolution for a finitely presented group whose presentation is known to have the property that its associated 2-complex is aspherical.

    gap> F:=FreeGroup(3);;x:=F.1;;y:=F.2;;z:=F.3;;
    gap> rels:=[x*y*x*(y*x*y)^-1, y*z*y*(z*y*z)^-1, z*x*z*(x*z*x)^-1];;
    gap> G:=F/rels;;
    gap> R:=ResolutionAsphericalPresentation(G,10);
    Resolution of length 10 in characteristic 0 for <fp group on the generators
    [ f1, f2, f3 ]> .
    No contracting homotopy available.

    gap> Size(R);
    [ 6, 18, 0, 0, 0, 0, 0, 0, 0, 0 ]

The following commands create a resolution for a graph of groups corresponding to the amalgamated product G=H∗_AK where H=S_5 is the symmetric group of degree 5, K=S_4 is the symmetric group of degree 4 and the common subgroup is A=S_3.

    gap> S5:=SymmetricGroup(5);SetName(S5,"S5");;
    Sym( [ 1 .. 5 ] )
    gap> S4:=SymmetricGroup(4);SetName(S4,"S4");;
    Sym( [ 1 .. 4 ] )
    gap> A:=SymmetricGroup(3);SetName(A,"S3");;
    Sym( [ 1 .. 3 ] )
    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,8);;
    gap> Size(R);
    [ 16, 68, 162, 302, 480, 627, 869, 1290 ]

#### 11.22 Resolutions for FG-modules

Let F= F_p be the field of p elements and let M be some FG-module for G a finite p-group. We might wish to construct a free FG-resolution for M. We can handle this by constructing a short exact sequence

DM ↣ P ↠ M

in which P is free (or projective). Then any resolution of DM yields a resolution of M and we can represent DM as a submodule of P. We refer to DM as the desuspension of M. Consider for instance G=Syl_2(GL(4,2)) and F= F_2. The matrix group G acts via matrix multiplication on M= F^4. The following example constructs a free FG-resolution for M.

    gap> G:=GL(4,2);;
    gap> S:=SylowSubgroup(G,2);;
    gap> M:=GModuleByMats(GeneratorsOfGroup(S),GF(2));;
    gap> DM:=DesuspensionMtxModule(M);;
    gap> R:=ResolutionFpGModule(DM,20);
    Resolution of length 20 in characteristic 2 for <matrix group of
    size 64 with 3 generators> .

    gap> List([0..20],R!.dimension);
    [ 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136,
    153, 171, 190, 210, 231, 253 ]

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