  
  [1X7 [33X[0;0YOther Functions in the [5XPERMUT[105X[101X[1X Package[133X[101X
  
  [33X[0;0YIn  this  chapter we define some miscellaneous functions which have appeared
  in  the context of permutability, or some functions which have been used for
  some of the functions of the package.[133X
  
  
  [1X7.1 [33X[0;0YFunctions[133X[101X
  
  [1X7.1-1 AllSubnormalSubgroups[101X
  
  [33X[1;0Y[29X[2XAllSubnormalSubgroups[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YThis  function  computes  all  subnormal  subgroups of [3XG[103X. The method used to
  obtain this list consists in beginning with the list of all normal subgroups
  of  [3XG[103X  and by adding all normal subgroups of the subgroups in the list until
  no  new  subnormal  subgroups  appear.  This  computes  the complete list of
  subgroups,  not  only  a  representative  of  each  conjugacy class as other
  functions do.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SymmetricGroup(4);[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XAllSubnormalSubgroups(g);[127X[104X
    [4X[28X[ Sym( [ 1 .. 4 ] ), Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,4)[128X[104X
    [4X[28X  (2,3), (1,3)(2,4) ]), Group(()), Group([ (1,3)(2,4) ]), Group([ (1,2)[128X[104X
    [4X[28X  (3,4) ]), Group([ (1,4)(2,3) ]) ][128X[104X
  [4X[32X[104X
  
  [1X7.1-2 PrimesDividingSize[101X
  
  [33X[1;0Y[29X[2XPrimesDividingSize[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YThis  attribute gives a list of primes dividing the size of the finite group
  [3XG[103X, without repetitions. Its code has been borrowed from the [5XGAP[105X manual.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SymmetricGroup(4);[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XPrimesDividingSize(g);[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
  [4X[32X[104X
  
  [1X7.1-3 SylowSubgroups[101X
  
  [33X[1;0Y[29X[2XSylowSubgroups[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YThis attribute returns a list composed by one Sylow subgroup for every prime
  dividing  the  size of [3XG[103X. If [3XG[103X is soluble, then it returns a Sylow system or
  Sylow   basis  of  [3XG[103X  by  means  of  the  function  [2XSylowSystem[102X  ([14XReference:
  SylowSystem[114X) (a set composed of a Sylow subgroup for each prime dividing the
  order of [3XG[103X permuting in pairs).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SymmetricGroup(4);[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XSylowSubgroups(g);[127X[104X
    [4X[28X[ Group([ (1,2), (3,4), (1,3)(2,4) ]), Group([ (1,2,3) ]) ][128X[104X
    [4X[25Xgap>[125X [27Xs5:=SymmetricGroup(5);[127X[104X
    [4X[28XSym( [ 1 .. 5 ] )[128X[104X
    [4X[25Xgap>[125X [27XSylowSubgroups(s5);[127X[104X
    [4X[28X[ Group([ (1,2), (3,4), (1,3)(2,4) ]), Group([ (1,2,3) ]), Group([ (1,2,3,4,[128X[104X
    [4X[28X   5) ]) ][128X[104X
  [4X[32X[104X
  
  [1X7.1-4 IsSCGroup[101X
  
  [33X[1;0Y[29X[2XIsSCGroup[102X( [3XG[103X ) [32X property[133X
  
  [33X[0;0YThis property is [9Xtrue[109X if [3XG[103X is an SC-group, and [9Xfalse[109X otherwise. A group [3XG[103X is
  an SC-group if all its chief factors are simple. Note that a soluble group [3XG[103X
  is  an  SC-group  if and only if [3XG[103X is supersoluble. The method used to check
  this  property uses the chief series if its is available or the group is not
  soluble.[133X
  
  [33X[0;0YSince  the  methods for insoluble groups might on the computation of a chief
  series  with  the  function [2XChiefSeries[102X ([14XReference: ChiefSeries[114X), they might
  not be available if the group is not given as a permutation group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SymmetricGroup(4);[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsSCGroup(g);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xg:=GL(2,5);[127X[104X
    [4X[28XGL(2,5)[128X[104X
    [4X[25Xgap>[125X [27XIsSCGroup(g);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.1-5 IsSylowTowerGroup[101X
  
  [33X[1;0Y[29X[2XIsSylowTowerGroup[102X( [3XG[103X ) [32X property[133X
  
  [33X[0;0YThis  property  takes  the value [9Xtrue[109X if [22XG[122X has a Sylow tower of supersoluble
  type, and [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 [27XIsSylowTowerGroup(g);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xg:=SmallGroup(75,1);[127X[104X
    [4X[28X<pc group of size 75 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSylowTowerGroup(g);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.1-6 Permutizer[101X
  
  [33X[1;0Y[29X[2XPermutizer[102X( [3XG[103X, [3XU[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPermutiser[102X( [3XG[103X, [3XU[103X ) [32X function[133X
  
  [33X[0;0YThe permutiser of a subgroup [3XU[103X of a group [3XG[103X is the subgroup generated by all
  cyclic  subgroups  of  [3XG[103X  which  permute with [3XU[103X. If [3XU[103X is permutable in [3XG[103X (in
  particular, if [3XU[103X is normal in [3XG[103X), then its permutizer coincides with [3XG[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SymmetricGroup(4);[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XPermutizer(g,Subgroup(g,[(1,2,3)]));[127X[104X
    [4X[28XGroup([ (1,2,3), (2,3) ])[128X[104X
    [4X[25Xgap>[125X [27XSize(last);[127X[104X
    [4X[28X6[128X[104X
  [4X[32X[104X
  
  [1X7.1-7 AllGeneratorsCyclicPGroup[101X
  
  [33X[1;0Y[29X[2XAllGeneratorsCyclicPGroup[102X( [3Xg[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis  auxiliary  function  returns  the list of all generators of the cyclic
  [22Xp[122X-group  generated  by the [22Xp[122X-element [22Xg[122X. Here [22Xp[122X is a prime number. Since this
  function  is  not  intended to be used in interactive mode, no check is done
  that the argument is a [22Xp[122X-element.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAllGeneratorsCyclicPGroup((1,2,3,4,5,6,7,8,9),3);[127X[104X
    [4X[28X[ (1,2,3,4,5,6,7,8,9), (1,3,5,7,9,2,4,6,8), (1,5,9,4,8,3,7,2,6),[128X[104X
    [4X[28X  (1,6,2,7,3,8,4,9,5), (1,8,6,4,2,9,7,5,3), (1,9,8,7,6,5,4,3,2) ][128X[104X
  [4X[32X[104X
  
