  
  [1X3 [33X[0;0YPermutability of Subgroups in Finite Groups[133X[101X
  
  [33X[0;0YThis  chapter  describes  functions to check permutability of subgroups in a
  given  group.  First  we  present  a  function  to  check whether a subgroup
  permutes  with  another  one,  then  we  present functions to test whether a
  subgroup  permutes  with  the  members  of  a given family of subgroups, and
  finally  we  introduce  some  other subgroup embedding properties related to
  permutability.[133X
  
  
  [1X3.1 [33X[0;0YPermutability functions[133X[101X
  
  [1X3.1-1 ArePermutableSubgroups[101X
  
  [33X[1;0Y[29X[2XArePermutableSubgroups[102X( [[3XG[103X, ][3XU[103X, [3XV[103X ) [32X function[133X
  
  [33X[0;0YThis  function returns [9Xtrue[109X if [3XU[103X and [3XV[103X permute in [3XG[103X. The groups [3XU[103X and [3XV[103X must
  be  subgroups  of  [3XG[103X.  The  subgroups  [22XU[122X and [22XV[122X [13Xpermute[113X when [22XUV = VU[122X. This is
  equivalent to affirming that [22XUV[122X is a subgroup of [22XG[122X.[133X
  
  [33X[0;0YThis  is  done  by checking that the order of [22X⟨ U, V ⟩[122X is the order of their
  Frobenius product [22XUV[122X, that is, [22X|U||V|/|U ∩ V|[122X. Hence the performance of this
  function depends strongly on the existence of good algorithms to compute the
  intersection  of  two  subgroups  and,  of  course, the order of a subgroup.
  Shorthands  are provided for the cases in which one of [3XU[103X and [3XV[103X is a subgroup
  of the other one or [3XU[103X or [3XV[103X are permutable in a common supergroup.[133X
  
  [33X[0;0YIn  the  version  with  two  arguments, [3XU[103X and [3XV[103X must have a common parent or
  [10XClosureGroup(  U, V )[110X (see [2XClosureGroup[102X ([14XReference: ClosureGroup[114X)) is called
  to construct a common supergroup for [3XU[103X and [3XV[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SymmetricGroup(4);[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27Xa:=Subgroup(g,[(1,2)(3,4)]);[127X[104X
    [4X[28XGroup([ (1,2)(3,4) ])[128X[104X
    [4X[25Xgap>[125X [27Xb:=Subgroup(g,[(1,2,3)]);[127X[104X
    [4X[28XGroup([ (1,2,3) ])[128X[104X
    [4X[25Xgap>[125X [27Xc:=Subgroup(g,[(1,2)]);[127X[104X
    [4X[28XGroup([ (1,2) ])[128X[104X
    [4X[25Xgap>[125X [27XArePermutableSubgroups(g,a,b);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XArePermutableSubgroups(g,a,c);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XArePermutableSubgroups(g,b,c);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XArePermutableSubgroups(b,c);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XArePermutableSubgroups(b,a);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YEmbedding properties related to permutability[133X[101X
  
  [33X[0;0YIn the following we describe some functions which allow us to test whether a
  subgroup  permutes  with  the  members of some families of subgroups. We pay
  special  attention  to the families of all subgroups and all Sylow subgroups
  of  the  group.  In some cases, we have introduced some [21XOne[121X functions, which
  give  an  element  or  a subgroup in the relevant family of subgroups of the
  group which shows that the given property fails, or [9Xfail[109X otherwise.[133X
  
  [1X3.2-1 PermutMaxTries[101X
  
  [33X[1;0Y[29X[2XPermutMaxTries[102X[32X global variable[133X
  
  [33X[0;0YThis   variable   contains   the   maximum  number  of  random  attempts  of
  permutability  checks  before  trying  general  deterministic methods in the
  functions [2XIsPermutable[102X ([14X3.2-2[114X) and [2XIsIwasawaSylow[102X ([14X5.3-5[114X). Its default value
  is [22X10[122X.[133X
  
  [1X3.2-2 IsPermutable[101X
  
  [33X[1;0Y[29X[2XIsPermutable[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsPermutableInParent[102X( [3XH[103X ) [32X property[133X
  
  [33X[0;0YThis  property  returns [9Xtrue[109X if the subgroup [3XH[103X is permutable in [3XG[103X, otherwise
  it  returns  [9Xfalse[109X. We say that a subgroup [22XH[122X of a group [22XG[122X is [13Xpermutable[113X in [22XG[122X
  if [22XH[122X permutes with all subgroups of [22XG[122X.[133X
  
  [33X[0;0YIf  the  attribute [2XOneSubgroupNotPermutingWithInParent[102X ([14X3.2-3[114X) has been set,
  it  is  used if possible. Otherwise, the algorithm checks looks for a cyclic
  subgroup  not  permuting  with  [3XH[103X.  The  number  of such cyclic subgroups is
  controlled  by  the  variable  [2XPermutMaxTries[102X  ([14X3.2-1[114X), by default, [22X10[122X. If [3XH[103X
  permutes  with  all  these subgroups, then the algorithm checks whether [3XH[103X is
  hypercentrally embedded in [3XG[103X and that the Sylow [22Xp[122X-subgroups of [22XH/H_G[122X permute
  with  all cyclic [22Xp[122X-subgroups of [22XG/H_G[122X for each prime [22Xp[122X dividing the order of
  [22XG/H_G[122X. This is a sufficient condition for permutability.[133X
  
  [1X3.2-3 OneSubgroupNotPermutingWith[101X
  
  [33X[1;0Y[29X[2XOneSubgroupNotPermutingWith[102X( [3XG[103X, [3XH[103X ) [32X function[133X
  [33X[1;0Y[29X[2XOneSubgroupNotPermutingWithInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YThis  attribute  finds a cyclic subgroup of [3XG[103X which does not permute with [3XH[103X,
  that is, a subgroup which shows that [3XH[103X is not permutable in [3XG[103X. Recall that a
  subgroup  [22XH[122X of a group [22XG[122X is [13Xpermutable[113X in [22XG[122X if [22XH[122X permutes with all subgroups
  of [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SymmetricGroup(4);[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27Xa:=Subgroup(g,[(1,2)(3,4)]);[127X[104X
    [4X[28XGroup([ (1,2)(3,4) ])[128X[104X
    [4X[25Xgap>[125X [27Xb:=Subgroup(g,[(1,2,3)]);[127X[104X
    [4X[28XGroup([ (1,2,3) ])[128X[104X
    [4X[25Xgap>[125X [27Xc:=Subgroup(g,[(1,2)]);[127X[104X
    [4X[28XGroup([ (1,2) ])[128X[104X
    [4X[25Xgap>[125X [27XIsPermutable(g,a);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsPermutable(g,b);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsPermutable(g,c);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XOneSubgroupNotPermutingWith(g,b);[127X[104X
    [4X[28XGroup([ (1,3,4) ])[128X[104X
    [4X[25Xgap>[125X [27Xv:=Subgroup(g,[(1,2)(3,4),(1,3)(2,4)]);[127X[104X
    [4X[28XGroup([ (1,2)(3,4), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XOneSubgroupNotPermutingWith(g,v);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XIsPermutable(g,b);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsPermutable(g,v);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SmallGroup(16,6);[127X[104X
    [4X[28X<pc group of size 16 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27Xh:=Subgroup(g,[g.2]);[127X[104X
    [4X[28XGroup([ f2 ])[128X[104X
    [4X[25Xgap>[125X [27XIsNormal(g,h);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsPermutable(g,h);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSometimes  one  does not require a subgroup to permute with all subgroups of
  the  group,  but  only with a selected family of subgroups of the group. The
  general case is the following.[133X
  
  [1X3.2-4 IsFPermutable[101X
  
  [33X[1;0Y[29X[2XIsFPermutable[102X( [3XG[103X, [3XH[103X, [3Xf[103X ) [32X function[133X
  
  [33X[0;0YIn  this function, [3XH[103X is a subgroup of [3XG[103X and [3Xf[103X must be a list of subgroups of
  [3XG[103X. It returns [9Xtrue[109X if [3XH[103X permutes with all members of [3Xf[103X and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YThis  function uses the function [2XOneFSubgroupNotPermutingWith[102X ([14X3.2-5[114X). Hence
  it   tries   to   use   the   values  of  [2XIsPermutableInParent[102X  ([14X3.2-2[114X)  and
  [2XOneSubgroupNotPermutingWithInParent[102X ([14X3.2-3[114X) if one of them is set, and if it
  returns [9Xfalse[109X it tries to set the values of [2XIsPermutableInParent[102X ([14X3.2-2[114X) and
  [2XOneSubgroupNotPermutingWithInParent[102X ([14X3.2-3[114X).[133X
  
  [1X3.2-5 OneFSubgroupNotPermutingWith[101X
  
  [33X[1;0Y[29X[2XOneFSubgroupNotPermutingWith[102X( [3XG[103X, [3XH[103X, [3Xf[103X ) [32X operation[133X
  
  [33X[0;0YIn this operation, [3XH[103X is a subgroup of [3XG[103X and [3Xf[103X must be a list of subgroups of
  [3XG[103X.  It  returns  a  subgroup  in  [3Xf[103X  not permuting with [3XH[103X if such a subgroup
  exists, and [9Xfail[109X otherwise.[133X
  
  [33X[0;0YThis  function  tries  to use the values of [2XIsPermutableInParent[102X ([14X3.2-2[114X) and
  [2XOneSubgroupNotPermutingWithInParent[102X  ([14X3.2-3[114X)  if  one  of them is set. If it
  returns [9Xfail[109X, then it tries to set the value of [2XIsPermutableInParent[102X ([14X3.2-2[114X)
  and [2XOneSubgroupNotPermutingWithInParent[102X ([14X3.2-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SymmetricGroup(4);[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27Xa:=Subgroup(g,[(1,2,3,4),(1,3)]);[127X[104X
    [4X[28XGroup([ (1,2,3,4), (1,3) ])[128X[104X
    [4X[25Xgap>[125X [27XSize(a);[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XOneFSubgroupNotPermutingWith(g,a,MaximalSubgroups(g));[127X[104X
    [4X[28XGroup([ (1,2), (3,4), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIsFPermutable(g,a,MaximalSubgroups(g));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XHasIsPermutableInParent(a);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPermutableInParent(a);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XHasOneSubgroupNotPermutingWithInParent(a);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOneSubgroupNotPermutingWithInParent(a);[127X[104X
    [4X[28XGroup([ (1,2), (3,4), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIsFPermutable(g,a,AllSubnormalSubgroups(g));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOneFSubgroupNotPermutingWith(g,a,AllSubnormalSubgroups(g));[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xsylows:=g->Union(List(SylowSubgroups(g),[127X[104X
    [4X[25X>[125X [27X          t->ConjugacyClassSubgroups(g,t)));[127X[104X
    [4X[28Xfunction( g ) ... end[128X[104X
    [4X[25Xgap>[125X [27XOneFSubgroupNotPermutingWith(g,a,sylows(g));[127X[104X
    [4X[28XGroup([ (3,4), (1,4)(2,3), (1,3)(2,4) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  functions  can  be  considered  as  particular  cases of the
  previous function for some subgroup embedding functors. However, they can be
  stored as [21Xin parent[121X attributes or properties and in some cases we have tried
  to give more efficient code.[133X
  
  [1X3.2-6 IsSPermutable[101X
  
  [33X[1;0Y[29X[2XIsSPermutable[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsSPermutableInParent[102X( [3XH[103X ) [32X property[133X
  
  [33X[0;0YThis  operation returns [9Xtrue[109X if a subgroup [3XH[103X of [3XG[103X is S-permutable in [3XG[103X, that
  is, [3XH[103X permutes with all Sylow subgroups of [3XG[103X, and returns [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SmallGroup(8,3);[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSPermutable(g,Subgroup(g,[g.1]));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPermutable(g,Subgroup(g,[g.1]));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X3.2-7 OneSylowSubgroupNotPermutingWith[101X
  
  [33X[1;0Y[29X[2XOneSylowSubgroupNotPermutingWith[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOneSylowSubgroupNotPermutingWithInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YThe  argument  [3XH[103X must be a subgroup of [3XG[103X. If [3XH[103X is S-permutable in [3XG[103X, then it
  returns  [9Xfail[109X.  Otherwise,  it  returns a Sylow subgroup of [3XG[103X which does not
  permute  with  [3XH[103X. We say that a subgroup [22XH[122X of a group [22XG[122X is S-permutable in [22XG[122X
  if [22XH[122X permutes with all Sylow subgroups of [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SymmetricGroup(4);;[127X[104X
    [4X[25Xgap>[125X [27Xa:=Subgroup(g,[(1,2)(3,4)]);;[127X[104X
    [4X[25Xgap>[125X [27XOneSylowSubgroupNotPermutingWith(g,a);[127X[104X
    [4X[28XGroup([ (2,4,3) ])[128X[104X
  [4X[32X[104X
  
  [1X3.2-8 IsSNPermutable[101X
  
  [33X[1;0Y[29X[2XIsSNPermutable[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsSNPermutableInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YThis  operation returns [9Xtrue[109X if [3XH[103X permutes with all system normalisers of [3XG[103X,
  and [9Xfalse[109X otherwise. Here [3XG[103X must be a soluble group and [3XH[103X must be a subgroup
  of [3XG[103X. If the function is applied to an insoluble group, it gives an error.[133X
  
  [1X3.2-9 OneSystemNormaliserNotPermutingWith[101X
  
  [33X[1;0Y[29X[2XOneSystemNormaliserNotPermutingWith[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOneSystemNormaliserNotPermutingWithInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XG[103X must be a soluble group and [3XH[103X must be a subgroup of [3XG[103X. If [3XH[103X permutes
  with  all  system  normalisers  of  [3XG[103X,  then  this  operation  returns [9Xfail[109X.
  Otherwise,  it  returns  a  system  normaliser  [22XD[122X  of [22XG[122X such that [22XH[122X does not
  permute with [22XD[122X. If the group [3XG[103X is not soluble, then it gives an error.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=Group((1,2,3),(4,5,6),(1,2));[127X[104X
    [4X[28XGroup([ (1,2,3), (4,5,6), (1,2) ])[128X[104X
    [4X[25Xgap>[125X [27Xa:=Subgroup(g,[(1,2,3)(4,5,6)]);[127X[104X
    [4X[28XGroup([ (1,2,3)(4,5,6) ])[128X[104X
    [4X[25Xgap>[125X [27XIsSNPermutable(g,a);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSPermutable(g,a);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next  functions  are  not  particular cases of [2XIsFPermutable[102X ([14X3.2-4[114X) or
  [2XOneFSubgroupNotPermutingWith[102X  ([14X3.2-5[114X),  but  we  include them in the package
  because  every  subgroup permuting with all its conjugates is subnormal (see
  [Fog97]).[133X
  
  [1X3.2-10 IsConjugatePermutable[101X
  
  [33X[1;0Y[29X[2XIsConjugatePermutable[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsConjugatePermutableInParent[102X( [3XH[103X ) [32X property[133X
  
  [33X[0;0YThis  operation  takes the value [9Xtrue[109X if [3XH[103X permutes with all its conjugates,
  and the value [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SymmetricGroup(4);[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27Xa:=Subgroup(g,[(1,2)(3,4)]);[127X[104X
    [4X[28XGroup([ (1,2)(3,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIsPermutable(g,a);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsConjugatePermutable(g,a);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.2-11 OneConjugateSubgroupNotPermutingWith[101X
  
  [33X[1;0Y[29X[2XOneConjugateSubgroupNotPermutingWith[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOneConjugateSubgroupNotPermutingWithInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YThis operation finds a conjugate subgroup of [3XH[103X which does not permute with [3XH[103X
  if  such a subgroup exists. If [3XH[103X permutes with all its conjugates, then this
  operation returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SmallGroup(16,7);[127X[104X
    [4X[28X<pc group of size 16 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27Xh:=Subgroup(g,[g.1*g.4]);[127X[104X
    [4X[28XGroup([ f1*f4 ])[128X[104X
    [4X[25Xgap>[125X [27XIsConjugatePermutable(g,h);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XOneConjugateSubgroupNotPermutingWith(g,h);[127X[104X
    [4X[28XGroup([ f1*f3 ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext we introduce some subgroup embedding functions related to permutability
  which  have  proved  to  be  useful  in  some  characterisations  of soluble
  T-groups,  PT-groups, and PST-groups. The [21XOne[121X functions return a value which
  proves that the corresponding subgroup embedding property is false.[133X
  
  [1X3.2-12 IsWeaklySPermutable[101X
  
  [33X[1;0Y[29X[2XIsWeaklySPermutable[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsWeaklySPermutableInParent[102X( [3XH[103X ) [32X property[133X
  
  [33X[0;0YThe value returned by this operation is [9Xtrue[109X when [3XH[103X is a [13Xweakly S-permutable[113X
  subgroup  of  [3XG[103X,  that is, [22XH[122X is S-permutable in [22X⟨ H, H^g ⟩[122X implies that [22XH[122X is
  S-permutable in [22X⟨ H, g ⟩[122X, and [9Xfalse[109X otherwise.[133X
  
  [1X3.2-13 OneElementShowingNotWeaklySPermutable[101X
  
  [33X[1;0Y[29X[2XOneElementShowingNotWeaklySPermutable[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOneElementShowingNotWeaklySPermutableInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YIf  [3XH[103X  is  a  weakly S-permutable subgroup of [3XG[103X, then this operation returns
  [9Xfail[109X.  Otherwise,  the  value returned by this operation is an element [22Xg ∈ G[122X
  such that [3XH[103X is S-permutable in [22X⟨ H, H^g ⟩[122X, but [22XH[122X is not S-permutable in [22X⟨ H,
  g  ⟩[122X.  A  subgroup  [22XH[122X of a group [22XG[122X is said to be weakly S-permutable if [22XH[122X is
  S-permutable in [22X⟨ H, H^g ⟩[122X implies that [22XH[122X is S-permutable in [22X⟨ H, g ⟩[122X.[133X
  
  [1X3.2-14 IsWeaklyPermutable[101X
  
  [33X[1;0Y[29X[2XIsWeaklyPermutable[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsWeaklyPermutableInParent[102X( [3XH[103X ) [32X property[133X
  
  [33X[0;0YThis  operation  returns  [9Xtrue[109X  if  [3XH[103X  is  weakly permutable in [3XG[103X, and [9Xfalse[109X
  otherwise.  A  subgroup  [22XH[122X  of  [22XG[122X is [13Xweakly permutable[113X if the fact that [22XH[122X is
  S-permutable in [22X⟨ H, H^g ⟩[122X, implies that [22XH[122X is S-permutable in [22X⟨ H, g ⟩[122X.[133X
  
  [1X3.2-15 OneElementShowingNotWeaklyPermutable[101X
  
  [33X[1;0Y[29X[2XOneElementShowingNotWeaklyPermutable[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOneElementShowingNotWeaklyPermutableInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YIf [3XH[103X is a weakly permutable subgroup of [3XG[103X, then this operation returns [9Xfail[109X.
  Otherwise,  the  value  returned  by this operation is an element [22Xg ∈ G[122X such
  that  [3XH[103X  is permutable in [22X⟨ H, H^g ⟩[122X, but [22XH[122X is not permutable in [22X⟨ H, g ⟩[122X. A
  subgroup  [22XH[122X  of a group [22XG[122X is said to be [13Xweakly permutable[113X if the fact that [22XH[122X
  is permutable in [22X⟨ H, H^g ⟩[122X implies that [22XH[122X is permutable in [22X⟨ H, g ⟩[122X.[133X
  
  [1X3.2-16 IsWeaklyNormal[101X
  
  [33X[1;0Y[29X[2XIsWeaklyNormal[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsWeaklyNormalInParent[102X( [3XH[103X ) [32X property[133X
  
  [33X[0;0YThis operation returns [9Xtrue[109X if [3XH[103X is weakly normal in [3XG[103X, and [9Xfalse[109X otherwise.
  A  subgroup  [22XH[122X  of  [22XG[122X  is  [13Xweakly  normal[113X whenever if [22XH^g ≤ N_G(H)[122X, then [22Xg ∈
  N_G(H)[122X.[133X
  
  [1X3.2-17 OneElementShowingNotWeaklyNormal[101X
  
  [33X[1;0Y[29X[2XOneElementShowingNotWeaklyNormal[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOneElementShowingNotWeaklyNormalInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YIf  [3XH[103X  is  a  weakly  normal subgruop of [3XG[103X, then this function returns [9Xfail[109X.
  Otherwise,  the  value  returned by this operation is an element [22Xg[122X such that
  [22XH^g≤ N_G(H)[122X is a subgroup of [22XN_G(H)[122X but [22Xg ∉ N_G(H)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=DihedralGroup(8);[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xa:=Subgroup(g,[g.1]);[127X[104X
    [4X[28XGroup([ f1 ])[128X[104X
    [4X[25Xgap>[125X [27XIsWeaklySPermutable(g,a);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsWeaklyPermutable(g,a);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xx:=OneElementShowingNotWeaklyPermutable(g,a);[127X[104X
    [4X[28Xf2[128X[104X
    [4X[25Xgap>[125X [27XIsSubgroup(Normalizer(g,a),ConjugateSubgroup(a,x));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xx in Normalizer(g,a);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X3.2-18 IsWithSubnormalizerCondition[101X
  
  [33X[1;0Y[29X[2XIsWithSubnormalizerCondition[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsWithSubnormalizerConditionInParent[102X( [3XH[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsWithSubnormaliserCondition[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsWithSubnormaliserConditionInParent[102X( [3XH[103X ) [32X property[133X
  
  [33X[0;0YThis  operation  returns  [9Xtrue[109X if the subgroup [22XH[122X satisfies the subnormaliser
  condition in [22XG[122X, and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YA  subgroup  [22XH[122X  is  said  to [13Xsatisfy the subnormaliser condition[113X in [22XG[122X if the
  condition  that [22XH[122X is subnormal in a subgroup [22XK[122X of [22XG[122X implies that [22XH[122X is normal
  in [22XK[122X.[133X
  
  [1X3.2-19 OneSubgroupInWhichSubnormalNotNormal[101X
  
  [33X[1;0Y[29X[2XOneSubgroupInWhichSubnormalNotNormal[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOneSubgroupInWhichSubnormalNotNormalInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YThis  function returns a subgroup [22XK[122X of [22XG[122X such that [22XH[122X is subnormal in [22XK[122X and [22XH[122X
  is not normal in [22XK[122X, if this subgroup exists; otherwise, it returns [9Xfail[109X.[133X
  
  [1X3.2-20 IsWithSubpermutizerCondition[101X
  
  [33X[1;0Y[29X[2XIsWithSubpermutizerCondition[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsWithSubpermutizerConditionInParent[102X( [3XH[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsWithSubpermutiserCondition[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsWithSubpermutiserConditionInParent[102X( [3XH[103X ) [32X property[133X
  
  [33X[0;0YThis  operation  returns  [9Xtrue[109X if the subgroup [22XH[122X satisfies the subpermutiser
  condition in [22XG[122X, and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YA  subgroup  [22XH[122X  is  said  to [13Xsatisfy the subpermutiser condition[113X in [22XG[122X if the
  condition  that  [22XH[122X  is  subnormal  in  a  subgroup  [22XK[122X of [22XG[122X implies that [22XH[122X is
  permutable in [22XK[122X.[133X
  
  [1X3.2-21 OneSubgroupInWhichSubnormalNotPermutable[101X
  
  [33X[1;0Y[29X[2XOneSubgroupInWhichSubnormalNotPermutable[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOneSubgroupInWhichSubnormalNotPermutableInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YThis  function returns a subgroup [22XK[122X of [22XG[122X such that [22XH[122X is subnormal in [22XK[122X and [22XH[122X
  is not permutable in [22XK[122X if this subgroup exists; otherwise it returns [9Xfail[109X.[133X
  
  [1X3.2-22 IsWithSSubpermutizerCondition[101X
  
  [33X[1;0Y[29X[2XIsWithSSubpermutizerCondition[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsWithSSubpermutizerConditionInParent[102X( [3XH[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsWithSSubpermutiserCondition[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsWithSSubpermutiserConditionInParent[102X( [3XH[103X ) [32X property[133X
  
  [33X[0;0YThis  operation returns [9Xtrue[109X if the subgroup [22XH[122X satisfies the S-subpermutiser
  condition in [22XG[122X, and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YA  subgroup  [22XH[122X  is said to [13Xsatisfy the S-subpermutiser condition[113X in [22XG[122X if the
  condition  that  [22XH[122X  is  subnormal  in  a  subgroup  [22XK[122X of [22XG[122X implies that [22XH[122X is
  S-permutable in [22XK[122X.[133X
  
  [1X3.2-23 OneSubgroupInWhichSubnormalNotSPermutable[101X
  
  [33X[1;0Y[29X[2XOneSubgroupInWhichSubnormalNotSPermutable[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOneSubgroupInWhichSubnormalNotSPermutableInParent[102X( [3XH[103X ) [32X attribute[133X
  
  [33X[0;0YThis  function returns a subgroup [22XK[122X of [22XG[122X such that [22XH[122X is subnormal in [22XK[122X and [22XH[122X
  is  not  S-permutable  in [22XK[122X  if such a subgroup exists; otherwise it returns
  [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SmallGroup(324,160);[127X[104X
    [4X[28X<pc group of size 324 with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27Xa:=Subgroup(g,[g.3,g.5]);[127X[104X
    [4X[28XGroup([ f3, f5 ])[128X[104X
    [4X[25Xgap>[125X [27XIsWithSubnormalizerCondition(g,a);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsWeaklyNormal(g,a);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsWeaklySPermutable(g,a);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xx:=OneElementShowingNotWeaklyNormal(g,a);[127X[104X
    [4X[28Xf1[128X[104X
    [4X[25Xgap>[125X [27XConjugateSubgroup(a,x)=a;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsSubset(Normalizer(g,a),ConjugateSubgroup(a,x));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
