  
  [1X9 [33X[0;0YSimplicial groups[133X[101X
  
  
  [1X9.1 [33X[0;0YCrossed modules[133X[101X
  
  [33X[0;0YThe following example concerns the crossed module[133X
  
  [33X[0;0Y[22X∂: G→ Aut(G), g↦ (x↦ gxg^-1)[122X[133X
  
  [33X[0;0Yassociated  to  the  dihedral  group  [22XG[122X  of  order  [22X16[122X.  This crossed module
  represents,  up to homotopy type, a connected space [22XX[122X with [22Xπ_iX=0[122X for [22Xige 3[122X,
  [22Xπ_2X=Z(G)[122X,  [22Xπ_1X  =  Aut(G)/Inn(G)[122X.  The  space  [22XX[122X can be represented, up to
  homotopy,  by  a  simplicial  group.  That  simplicial  group is used in the
  example to compute[133X
  
  [33X[0;0Y[22XH_1(X, Z)= Z_2 ⊕ Z_2[122X,[133X
  
  [33X[0;0Y[22XH_2(X, Z)= Z_2[122X,[133X
  
  [33X[0;0Y[22XH_3(X, Z)= Z_2 ⊕ Z_2 ⊕ Z_2[122X,[133X
  
  [33X[0;0Y[22XH_4(X, Z)= Z_2 ⊕ Z_2 ⊕ Z_2[122X,[133X
  
  [33X[0;0Y[22XH_5(X, Z)= Z_2 ⊕ Z_2 ⊕ Z_2 ⊕ Z_2⊕ Z_2⊕ Z_2[122X.[133X
  
  [33X[0;0YThe  simplicial group is obtained by viewing the crossed module as a crossed
  complex and using a nonabelian version of the Dold-Kan theorem.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XC:=AutomorphismGroupAsCatOneGroup(DihedralGroup(16));[127X[104X
    [4X[28XCat-1-group with underlying group Group( [128X[104X
    [4X[28X[ f1, f2, f3, f4, f5, f6, f7, f8, f9 ] ) . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(C);[127X[104X
    [4X[28X512[128X[104X
    [4X[25Xgap>[125X [27XQ:=QuasiIsomorph(C);[127X[104X
    [4X[28XCat-1-group with underlying group Group( [ f9, f8, f1, f2*f3, f5 ] ) . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(Q);[127X[104X
    [4X[28X32[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XN:=NerveOfCatOneGroup(Q,6);[127X[104X
    [4X[28XSimplicial group of length 6[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XK:=ChainComplexOfSimplicialGroup(N);[127X[104X
    [4X[28XChain complex of length 6 in characteristic 0 . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XHomology(K,1);[127X[104X
    [4X[28X[ 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(K,2);[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(K,3);[127X[104X
    [4X[28X[ 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(K,4);[127X[104X
    [4X[28X[ 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(K,5);[127X[104X
    [4X[28X[ 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X9.2 [33X[0;0YEilenberg-MacLane spaces[133X[101X
  
  [33X[0;0YThe  following  example concerns the Eilenberg-MacLane space [22XX=K( Z,3)[122X which
  is  a path-connected space with [22Xπ_3X= Z[122X, [22Xπ_iX=0[122X for [22X3ne ige 1[122X. This space is
  represented  by  a simplicial group, and perturbation techniques are used to
  compute[133X
  
  [33X[0;0Y[22XH_7(X, Z)= Z_3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:=AbelianPcpGroup([0]);;AbelianInvariants(A);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[25Xgap>[125X [27XK:=EilenbergMacLaneSimplicialGroup(A,3,8);[127X[104X
    [4X[28XSimplicial group of length 8[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XC:=ChainComplexOfSimplicialGroup(K);[127X[104X
    [4X[28XChain complex of length 8 in characteristic 0 . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XHomology(C,7);[127X[104X
    [4X[28X[ 3 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
