  
  [1X4 [33X[0;0YExample computations with almost crystallographic groups[133X[101X
  
  
  [1X4.1 [33X[0;0YExample computations I[133X[101X
  
  [33X[0;0YUsing the functions available for pcp groups in the [5XPolycyclic[105X package it is
  now  easy to redo some of the calculations of [Dek96]. As a first example we
  check  whether  the  groups  indicated  as  torsion free in [Dek96] are also
  determined  as  torsion free ones by [5XGAP[105X. In [Dek96] these almost Bieberbach
  groups  are  listed  as [21XAB-groups[121X. So for type [21X013[121X these are the groups with
  parameters  [22X(k,0,1,0,1,0)[122X  where  [22Xk[122X  is  an even integer. Let's look at some
  examples in [5XGAP[105X:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=AlmostCrystallographicPcpDim4("013",[8,0,1,0,1,0]);[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XIsTorsionFree(G);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XG:=AlmostCrystallographicPcpDim4("013",[9,0,1,0,1,0]);[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XIsTorsionFree(G);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFurther, there is also some cohomology information in the tables of [Dek96].
  In  fact,  the  groups  in this library were obtained as extensions [22XE[122X of the
  form[133X
  
  
  [24X[33X[0;6Y1 \rightarrow ℤ \rightarrow E \rightarrow Q \rightarrow 1[133X
  
  [124X
  
  [33X[0;0Ywhere, in the 4-dimensional case [22XQ = E/⟨ d ⟩[122X. The cohomology information for
  the particular example above shows that the groups determined by a parameter
  set  [22X(k_1,k_2,k_3,k_4,k_5,k_6)[122X  are  equivalent  as extensions to the groups
  determined  by the parameters [22X(k_1, k_2 mod 2, k_3 mod 2, k_4 mod 2, k_5 mod
  2, 0)[122X. This is also visible in finding torsion:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=AlmostCrystallographicPcpDim4("013",[10,0,2,0,1,0]);[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XIsTorsionFree(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XG:=AlmostCrystallographicPcpDim4("013",[10,0,3,0,1,9]);[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XIsTorsionFree(G);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X4.2 [33X[0;0YExample computations II[133X[101X
  
  [33X[0;0YThe  computation  of  cohomology  groups  played  an  important  role in the
  classification  of the almost Bieberbach groups in [Dek96]. Using [5XGAP[105X, it is
  now  possible  to  check  these  computations. As an example we consider the
  4-dimensional  almost  crystallographic  group  of  type  85  on page 202 of
  [Dek96].  This  group  [22XE[122X  has 6 generators. In the table, one also finds the
  information[133X
  
  
  [24X[33X[0;6YH^2(Q,ℤ) = ℤ \oplus (ℤ_2)^2 \oplus ℤ_4[133X
  
  [124X
  
  [33X[0;0Yfor [22XQ=E/⟨ d ⟩[122X as above. Moreover, the [22XQ[122X-module [22Xℤ[122X is in fact the group [22X⟨ d ⟩[122X,
  where the [22XQ[122X-action comes from conjugation inside [22XE[122X. In the case of groups of
  type  85, [22Xℤ[122X is a trivial [22XQ[122X-module. The following example demonstrates how to
  (re)compute this two-cohomology group [22XH^2(Q,ℤ)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=AlmostCrystallographicPcpGroup(4, "085", false);[127X[104X
    [4X[28XPcp-group with orders [ 2, 4, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XGroupGeneratedByd:=Subgroup(G, [G.6] );[127X[104X
    [4X[28XPcp-group with orders [ 0 ][128X[104X
    [4X[25Xgap>[125X [27XQ:=G/GroupGeneratedByd;[127X[104X
    [4X[28XPcp-group with orders [ 2, 4, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xaction:=List( Pcp(Q), x -> [[1]] );[127X[104X
    [4X[28X[ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XC:=CRRecordByMats( Q, action);;[127X[104X
    [4X[25Xgap>[125X [27XTwoCohomologyCR( C ).factor.rels;[127X[104X
    [4X[28X[ 2, 2, 4, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  last  line  gives  us  the abelian invariants of the second cohomology
  group [22XH^2(Q,ℤ)[122X. So we should read this line as[133X
  
  
  [24X[33X[0;6YH^2(Q,ℤ) = ℤ_2 \oplus ℤ_2 \oplus ℤ_4 \oplus ℤ[133X
  
  [124X
  
  [33X[0;0Ywhich indeed coincides with the information in [Dek96].[133X
  
  
  [1X4.3 [33X[0;0YExample computations III[133X[101X
  
  [33X[0;0YAs  another  application of the capabilities of the combination of [5XAClib[105X and
  [5XPolycyclic[105X we check some computations of [DM01].[133X
  
  [33X[0;0YSection  5  of  the  paper [DM01] is completely devoted to an example of the
  computation of the [22XP[122X-localization of a virtually nilpotent group, where [22XP[122X is
  a  set  of primes. Although it is not our intention to develop the theory of
  [22XP[122X-localization  of  groups  at this place, let us summarize some of the main
  results concerning this topic here.[133X
  
  [33X[0;0YFor  a  set  of primes [22XP[122X, we say that [22Xn ∈ P[122X if and only if [22Xn[122X is a product of
  primes  in  [22XP[122X. A group [22XG[122X is said to be [22XP[122X-local if and only if the map [22Xμ_n:G→
  G:  g  ↦  g^n[122X is bijective for all [22Xn ∈ P'[122X, where [22XP'[122X is the set of all primes
  not  in  [22XP[122X. The [22XP[122X-localization of a group [22XG[122X, is a [22XP[122X-local group [22XG_P[122X together
  with  a  morphism [22Xα :G → G_P[122X which satisfy the following universal property:
  For  each  [22XP[122X-local  group [22XL[122X and any morphism [22Xφ: G → L[122X, there exists a unique
  morphism [22Xψ:G_P → L[122X, such that [22Xψ ∘ α = φ[122X.[133X
  
  [33X[0;0YThis  concept  of  localization  is well developed for finite groups and for
  nilpotent  groups.  For  a finite group [22XG[122X, the [22XP[122X-localization is the largest
  quotient  of  [22XG[122X,  having  no  elements  with  an  order belonging to [22XP'[122X (the
  morphism [22Xα[122X, mentioned above is the natural projection).[133X
  
  [33X[0;0YIn  [DM01]  a  contribution  is  made  towards the localization of virtually
  nilpotent  groups.  The theory developed in the paper is then illustrated in
  the  last  section  of  the  paper  by  means  of  the  computation  of  the
  [22XP[122X-localization  of  an  almost crystallographic group. For their example the
  authors  have  chosen  an almost crystallographic group [22XG[122X of dimension 3 and
  type  17.  For  the set of parameters [22X(k_1,k_2,k_3,k_4)[122X they have considered
  all cases of the form [22X(k_1,k_2,k_3,k_4)=(2,0,0,k_4)[122X.[133X
  
  [33X[0;0YHere we will check their computations in two cases [22Xk_4=0[122X and [22Xk_4=1[122X using the
  set  of  primes  [22XP={2}[122X.  The holonomy group of these almost crystallographic
  groups  [22XG[122X  is  the dihedral group [22Xcal D_6[122X of order 12. Thus there is a short
  exact sequence of the form[133X
  
  
  [24X[33X[0;6Y1 \rightarrow Fitt(G) \rightarrow G \rightarrow {\cal D}_6 \rightarrow 1.[133X
  
  [124X
  
  [33X[0;0YAs  a  first  step in their computation, Descheemaeker and Malfait determine
  the  group  [22XI_P'cal D_6[122X, which is the unique subgroup of order 3 in [22Xcal D_6[122X.
  One of the main objects in [DM01] is the group [22XK=p^-1 (I_P'cal D_6)[122X, where [22Xp[122X
  is the natural projection of [22XG[122X onto its holonomy group. It is known that the
  [22XP[122X-localization  of  [22XG[122X  coincides  with the [22XP[122X-localization of [22XG/γ_3(K)[122X, where
  [22Xγ_3(K)[122X  is  the  third term in the lower central series of [22XK[122X. As [22XG/γ_3(K)[122X is
  finite  in this example, we exactly know what this [22XP[122X-localization is. Let us
  now show how [5XGAP[105X can be used to compute this [22XP[122X-localization in two cases:[133X
  
  [33X[0;0YFirst case: The parameters are [22X(k_1,k_2,k_3,k_4)=(2,0,0,0)[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := AlmostCrystallographicPcpGroup(3, 17, [2,0,0,0] );[127X[104X
    [4X[28XPcp-group with orders [ 2, 6, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xprojection := NaturalHomomorphismOnHolonomyGroup( G );[127X[104X
    [4X[28X[ g1, g2, g3, g4, g5 ] -> [ g1, g2, id, id, id ][128X[104X
    [4X[25Xgap>[125X [27XF := HolonomyGroup( G );[127X[104X
    [4X[28XPcp-group with orders [ 2, 6 ][128X[104X
    [4X[25Xgap>[125X [27XIPprimeD6 := Subgroup( F , [F.2^2] );[127X[104X
    [4X[28XPcp-group with orders [ 3 ][128X[104X
    [4X[25Xgap>[125X [27XK := PreImage( projection, IPprimeD6 );[127X[104X
    [4X[28XPcp-group with orders [ 3, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XPrintPcpPresentation( K );[127X[104X
    [4X[28Xg1^3 = id [128X[104X
    [4X[28Xg2 ^ g1 = g2^-1 * g3^-1[128X[104X
    [4X[28Xg3 ^ g1 = g2 * g4^2[128X[104X
    [4X[28Xg3 ^ g2 = g3 * g4^2[128X[104X
    [4X[28Xg3 ^ g2^-1 = g3 * g4^-2[128X[104X
    [4X[25Xgap>[125X [27XGamma3K := CommutatorSubgroup( K, CommutatorSubgroup( K, K ));[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xquotient := G/Gamma3K;[127X[104X
    [4X[28XPcp-group with orders [ 2, 6, 3, 3, 2 ][128X[104X
    [4X[25Xgap>[125X [27XS := SylowSubgroup( quotient, 3);[127X[104X
    [4X[28XPcp-group with orders [ 3, 3, 3 ][128X[104X
    [4X[25Xgap>[125X [27XN := NormalClosure( quotient, S);[127X[104X
    [4X[28XPcp-group with orders [ 3, 3, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xlocalization := quotient/N;[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XPrintPcpPresentation( localization );[127X[104X
    [4X[28Xg1^2 = id [128X[104X
    [4X[28Xg2^2 = id [128X[104X
    [4X[28Xg3^2 = id [128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis shows that [22XG_P≅ ℤ_2^3[122X.[133X
  
  [33X[0;0YSecond case: The parameters are [22X(k_1,k_2,k_3,k_4)=(2,0,0,1)[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := AlmostCrystallographicPcpGroup(3, 17, [2,0,0,1]);;[127X[104X
    [4X[25Xgap>[125X [27Xprojection := NaturalHomomorphismOnHolonomyGroup( G );;[127X[104X
    [4X[25Xgap>[125X [27XF := HolonomyGroup( G );;[127X[104X
    [4X[25Xgap>[125X [27XIPprimeD6 := Subgroup( F , [F.2^2] );;[127X[104X
    [4X[25Xgap>[125X [27XK := PreImage( projection, IPprimeD6 );;[127X[104X
    [4X[25Xgap>[125X [27XGamma3K := CommutatorSubgroup( K, CommutatorSubgroup( K, K ));;[127X[104X
    [4X[25Xgap>[125X [27Xquotient := G/Gamma3K;;[127X[104X
    [4X[25Xgap>[125X [27XS := SylowSubgroup( quotient, 3);;[127X[104X
    [4X[25Xgap>[125X [27XN := NormalClosure( quotient, S);;[127X[104X
    [4X[25Xgap>[125X [27Xlocalization := quotient/N;[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XPrintPcpPresentation( localization );[127X[104X
    [4X[28Xg1^2 = id [128X[104X
    [4X[28Xg2^2 = g3[128X[104X
    [4X[28Xg3^2 = id [128X[104X
    [4X[28Xg2 ^ g1 = g2 * g3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn this case, we see that [22XG_P=cal D_4[122X.[133X
  
  [33X[0;0YThe  reader  can  check  that  these results coincide with those obtained in
  [DM01].  Note  also that we used a somewhat different scheme to compute this
  localization  than the one used in [DM01]. We invite the reader to check the
  same computations, tracing exactly the steps made in [DM01].[133X
  
  [33X[0;0Y [133X
  
