  
  [1X5 [33X[0;0YGroup theoretic computations[133X[101X
  
  
  [1X5.1 [33X[0;0YThird homotopy group of a supsension of an Eilenberg-MacLane space[133X[101X
  
  [33X[0;0YThe following example uses the nonabelian tensor square of groups to compute
  the third homotopy group[133X
  
  [33X[0;0Y[22Xπ_3(S(K(G,1))) = Z^30[122X[133X
  
  [33X[0;0Yof  the  suspension  of  the  Eigenberg-MacLane  space [22XK(G,1)[122X for [22XG[122X the free
  nilpotent group of class [22X2[122X on four generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup(4);;G:=NilpotentQuotient(F,2);;[127X[104X
    [4X[25Xgap>[125X [27XThirdHomotopyGroupOfSuspensionB(G);[127X[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 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X5.2 [33X[0;0YRepresentations of knot quandles[133X[101X
  
  [33X[0;0YThe  following example constructs the finitely presented quandles associated
  to  the granny knot and square knot, and then computes the number of quandle
  homomorphisms  from  these  two  finitely  prresented  quandles to the [22X17[122X-th
  quandle  in  [12XHAP[112X's  library of connected quandles of order [22X24[122X. The number of
  homomorphisms  differs  between  the  two  cases.  The computation therefore
  establishes   that  the  complement  in  [22XR^3[122X  of  the  granny  knot  is  not
  homeomorphic to the complement of the square knot.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQ:=ConnectedQuandle(24,17,"import");;[127X[104X
    [4X[25Xgap>[125X [27XK:=PureCubicalKnot(3,1);;[127X[104X
    [4X[25Xgap>[125X [27XL:=ReflectedCubicalKnot(K);;[127X[104X
    [4X[25Xgap>[125X [27Xsquare:=KnotSum(K,L);;[127X[104X
    [4X[25Xgap>[125X [27Xgranny:=KnotSum(K,K);;[127X[104X
    [4X[25Xgap>[125X [27Xgcsquare:=GaussCodeOfPureCubicalKnot(square);;[127X[104X
    [4X[25Xgap>[125X [27Xgcgranny:=GaussCodeOfPureCubicalKnot(granny);;[127X[104X
    [4X[25Xgap>[125X [27XQsquare:=PresentationKnotQuandle(gcsquare);;[127X[104X
    [4X[25Xgap>[125X [27XQgranny:=PresentationKnotQuandle(gcgranny);;[127X[104X
    [4X[25Xgap>[125X [27XNumberOfHomomorphisms(Qsquare,Q);[127X[104X
    [4X[28X408[128X[104X
    [4X[25Xgap>[125X [27XNumberOfHomomorphisms(Qgranny,Q);[127X[104X
    [4X[28X24[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X5.3 [33X[0;0YAspherical [22X2[122X[101X[1X-complexes[133X[101X
  
  [33X[0;0YThe  following  example  uses  Polymake's  linear  programming  routines  to
  establish  that  the [22X2[122X-complex associated to the group presentation [22X<x,y,z :
  xyx=yxy,   yzy=zyz,  xzx=zxz>[122X  is  aspherical  (that  is,  has  contractible
  universal  cover). The presentation is Tietze equivalent to the presentation
  used  in the computer code, and the associated [22X2[122X-complexes are thus homotopy
  equivalent.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup(6);;[127X[104X
    [4X[25Xgap>[125X [27Xx:=F.1;;y:=F.2;;z:=F.3;;a:=F.4;;b:=F.5;;c:=F.6;;[127X[104X
    [4X[25Xgap>[125X [27Xrels:=[a^-1*x*y, b^-1*y*z, c^-1*z*x, a*x*(y*a)^-1,[127X[104X
    [4X[25X>[125X [27X   b*y*(z*b)^-1, c*z*(x*c)^-1];;[127X[104X
    [4X[25Xgap>[125X [27XPrint(IsAspherical(F,rels),"\n");[127X[104X
    [4X[28XPresentation is aspherical.[128X[104X
    [4X[28X[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X5.4 [33X[0;0YBogomolov multiplier[133X[101X
  
  [33X[0;0YThe  Bogomolov  multiplier of a group is an isoclinism invariant. Using this
  property,  the following example shows that there are precisely three groups
  of  order  [22X243[122X with non-trivial Bogomolov multiplier. The groups in question
  are numbered 28, 29 and 30 in [12XGAP[112X's library of small groups of order [22X243[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=AllSmallGroups(3^5);;[127X[104X
    [4X[25Xgap>[125X [27XC:=IsoclinismClasses(L);;[127X[104X
    [4X[25Xgap>[125X [27Xfor c in C do[127X[104X
    [4X[25X>[125X [27Xif Length(BogomolovMultiplier(c[1]))>0 then[127X[104X
    [4X[25X>[125X [27XPrint(List(c,g->IdGroup(g)),"\n\n\n"); fi;[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[28X[ [ 243, 28 ], [ 243, 29 ], [ 243, 30 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
