  
  [1X5 [33X[0;0YLocal Functions in the [5XPERMUT[105X[101X[1X Package[133X[101X
  
  [33X[0;0YIn  the  study  of  permutability,  the usage of local characterisations has
  become  a  useful  tool  to describe the classes of T-groups, PT-groups, and
  PST-groups. In this chapter we present some local characterisations of these
  classes and some functions which allow to check whether or not a group given
  in [5XGAP[105X satisfies these conditions.[133X
  
  [33X[0;0YA  [13Xlocal[113X description of group-theoretical property consists of expressing it
  as  the  conjunction  of  some  properties  depending  on a prime [22Xp[122X, usually
  related to the behaviour of [22Xp[122X-elements, [22Xp[122X-subgroups, or [22Xp[122X-chief factors, for
  all primes [22Xp[122X.[133X
  
  
  [1X5.1 [33X[0;0YA Local Function for Supersolubility[133X[101X
  
  [33X[0;0YThe  [5XGAP[105X  library  does  not  contain  methods to check whether a group [22XG[122X is
  [22Xp[122X-supersoluble, where [22Xp[122X is a prime number. We include such a function in the
  [5XPERMUT[105X package.[133X
  
  [1X5.1-1 IsPSupersolvable[101X
  
  [33X[1;0Y[29X[2XIsPSupersolvable[102X( [3XG[103X, [3Xp[103X ) [32X function[133X
  [33X[1;0Y[29X[2XIsPSupersoluble[102X( [3XG[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis  function  returns  [9Xtrue[109X  if  the  group [3XG[103X is [22Xp[122X-supersoluble, and [9Xfalse[109X
  otherwise,  where  [3Xp[103X is a prime number. This function is not defined in [5XGAP[105X.
  The  method  we have implemented for finite groups includes checking whether
  the  group is supersoluble (in this case, it must return [9Xtrue[109X). If the group
  is  not  soluble,  it  computes  a chief series and checks whether all chief
  factors have order [22Xp[122X or have order not divisible by [22Xp[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=Group((1,2,3,4,5,6,7), (8,9,10,11,12,13,14), (15,16,17,18,19,20,21),[127X[104X
    [4X[25X>[125X [27X(22,23,24,25,26,27,28), (29,30,31,32,33,34,35),[127X[104X
    [4X[25X>[125X [27X(1,8,15,22,29)(2,9,16,23,30)(3,10,17,24,31)(4,11,18,25,32)(5,12,19,26,[127X[104X
    [4X[25X>[125X [27X    33)(6,13,20,27,34)(7,14,21,28,35),[127X[104X
    [4X[25X>[125X [27X(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)); #C7 wr S5[127X[104X
    [4X[28X<permutation group with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsPSupersolvable(g,7);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsPSupersolvable(g,11);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=DirectProduct(PSL(2,7),[127X[104X
    [4X[25X>[125X [27X    Group((1,2,3,4,5,6,7,8,9,10,11), (2,5,6,10,4)(3,9,11,8,7)));[127X[104X
    [4X[28XGroup([ (3,7,5)(4,8,6), (1,2,6)(3,4,8), (9,10,11,12,13,14,15,16,17,18,19),[128X[104X
    [4X[28X  (10,13,14,18,12)(11,17,19,16,15) ])[128X[104X
    [4X[25Xgap>[125X [27XIsPNilpotent(g,5);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPNilpotent(g,11);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsPSupersolvable(g,11);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPNilpotent(g,3);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X5.2 [33X[0;0YLocal functions for T-groups, PT-groups, and PST-groups[133X[101X
  
  [33X[0;0YThe  following  functions  correspond to local description of the classes of
  soluble  T-groups, PT-groups, and PST-groups. Most of the known useful local
  characterisations of these classes of groups can be seen to be equivalent to
  one  of them, either in the universe or all finite groups or in the universe
  of   all   finite  [22Xp[122X-soluble  groups.  By  a  local  characterisation  of  a
  group-theoretical  property  [22XR[122X  we mean a group-theoretical property [22XR_p[122X for
  each  prime  [22Xp[122X such that a group satisfies [22XR[122X if and only if it satisfies [22XR_p[122X
  for all primes [22Xp[122X.[133X
  
  [1X5.2-1 IsCp[101X
  
  [33X[1;0Y[29X[2XIsCp[102X( [3XG[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis  function returns [9Xtrue[109X if the group [3XG[103X satisfies the property [22XC_p[122X, where
  [22Xp[122X is a prime number, and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YA  group  [22XG[122X satisfies [22XC_p[122X when every subgroup [22XH[122X of a Sylow [22Xp[122X-subgroup [22XP[122X of [22XG[122X
  is  normal  in  the corresponding Sylow normaliser [22XN_G(P)[122X. This property was
  introduced  by  Robinson in [Rob68]. A group [22XG[122X is a soluble PST-group if and
  only if it satisfies [22XC_p[122X for all primes [22Xp[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=AlternatingGroup(5);[127X[104X
    [4X[28XAlt( [ 1 .. 5 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsCp(g,3);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCp(g,5);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCp(g,7);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCp(g,2);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xg:=SmallGroup(200,44); # semidirect product of Q8 with C5xC5[127X[104X
    [4X[28X<pc group of size 200 with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsCp(g,5);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsCp(g,2);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X5.2-2 IsXp[101X
  
  [33X[1;0Y[29X[2XIsXp[102X( [3XG[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis function returns [9Xtrue[109X if [3XG[103X satisfies [22XX_p[122X, where [22Xp[122X is a prime, and [9Xfalse[109X
  otherwise.[133X
  
  [33X[0;0YA group [22XG[122X satisfies [22XX_p[122X when for every subgroup [22XH[122X of a Sylow [22Xp[122X-subgroup [22XP[122X of
  [22XG[122X,  [22XH[122X  is  permutable  in [22XN_G(P)[122X. This property was introduced by Beidleman,
  Brewster,  and  Robinson  in [BBR99]. A group [22XG[122X is a soluble PT-group if and
  only if [22XG[122X satisfies [22XX_p[122X for all primes [22Xp[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SmallGroup(189,7);[127X[104X
    [4X[28X<pc group of size 189 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsXp(g,3);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsXp(g,7);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPTGroup(g);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCp(g,3);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsTGroup(g);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X5.2-3 IsYp[101X
  
  [33X[1;0Y[29X[2XIsYp[102X( [3XG[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis function returns [9Xtrue[109X if [3XG[103X satisfies [22XY_p[122X, where [22Xp[122X is a prime, and [9Xfalse[109X
  otherwise.[133X
  
  [33X[0;0YA group [22XG[122X satisfies [22XY_p[122X when for every two subgroups [22XH[122X and [22XK[122X with [22XH≤ K[122X, [22XH[122X is
  S-permutable  in [22XN_G(P)[122X. This property was introduced by Ballester-Bolinches
  and Esteban-Romero in [BBER02]. A group [22XG[122X is a soluble PST-group if and only
  if [22XG[122X satisfies [22XY_p[122X for all primes [22Xp[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=SmallGroup(200,43); # semidirect product of D8 with C5xC5[127X[104X
    [4X[28X<pc group of size 200 with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsCp(g,2);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsXp(g,2);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsYp(g,2);[127X[104X
    [4X[28Xtrue[128X[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 [27XIsYp(g,3);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsYp(g,2);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X5.3 [33X[0;0YAuxiliary Functions for T-groups, PT-groups, and PST-groups[133X[101X
  
  [33X[0;0YThe  following  functions  are  used  to  check  whether or not a group is a
  soluble T-group, PT-group, or PST-group.[133X
  
  [1X5.3-1 IsAbCp[101X
  
  [33X[1;0Y[29X[2XIsAbCp[102X( [3XG[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis  function returns [9Xtrue[109X if [3XG[103X has an abelian Sylow [3Xp[103X-subgroup [3Xp[103X such that
  every  subgroup  of  [22XP[122X  is  normal in the Sylow normaliser [22XN_G(P)[122X, and [9Xfalse[109X
  otherwise.[133X
  
  [33X[0;0YThis  function  is  used  to characterise soluble PST-groups: a group [22XG[122X is a
  soluble  PST-group  if  and  only if [22XG[122X satisfies [22XY_p[122X for all primes [22Xp[122X, and a
  group  [22XG[122X  satisfies  [22XY_p[122X if and only if [22XG[122X is [22Xp[122X-nilpotent or [22XG[122X has an abelian
  Sylow  [22Xp[122X-subgroup  and satisfies [22XC_p[122X. A group [22XG[122X satisfies [22XC_p[122X if and only if
  every  subgroup  of  a  Sylow  [22Xp[122X-subgroup  [22XP[122X  of  [22XG[122X  is  normal in the Sylow
  normaliser  [22XN_G(P)[122X.  Therefore this function checks whether [22XG[122X has an abelian
  Sylow [22Xp[122X-subgroup and [22XG[122X satisfies [22XC_p[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=AlternatingGroup(5);[127X[104X
    [4X[28XAlt( [ 1 .. 5 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsAbCp(g,5);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X5.3-2 IsDedekindSylow[101X
  
  [33X[1;0Y[29X[2XIsDedekindSylow[102X( [3XG[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis  function  returns [9Xtrue[109X if a Sylow [3Xp[103X-subgroup of [3XG[103X is Dedekind, else it
  returns [9Xfalse[109X.[133X
  
  [33X[0;0YA group [22XG[122X is Dedekind when every subgroup of [22XG[122X is normal. If [22Xp[122X is a prime, a
  Dedekind  [22Xp[122X-group (see for example 2.3.12 in [Sch94]) is abelian or a direct
  product  of a quaternion group of order [22X8[122X and an elementary abelian [22X2[122X-group.
  Obviously, a [22Xp[122X-group is Dedekind if and only if it is a T-group.[133X
  
  [33X[0;0YThe  algorithm  used  in this function to test whether a non-abelian [22X2[122X-group
  satisfies  this  condition  checks  that  the  Frattini  subgroup of a Sylow
  [22X2[122X-subgroup  [22XP[122X  of  [22XG[122X has order [22X2[122X and that the centre of [22XP[122X has exponent [22X2[122X and
  index  [22X4[122X. In this case, it computes the natural epimorphism from [22XP[122X to [22XP/Z(P)[122X
  and  it  checks  that  the  preimages  of the generators of [22XP/Z(P)[122X under the
  natural  epimorphism  have  order  [22X4[122X. If all these conditions hold, then the
  Sylow [22X2[122X-subgroup is Dedekind, otherwise it is not.[133X
  
  [33X[0;0YThis  function  tries  to set the property [2XIsTGroup[102X ([14X4.2-1[114X) to [9Xtrue[109X or [9Xfalse[109X
  for the Sylow [22Xp[122X-subgroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=DirectProduct(SmallGroup(8,4),CyclicGroup(5));[127X[104X
    [4X[28X<pc group of size 40 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsDedekindSylow(g,2);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X5.3-3 IwasawaTripleWithSubgroup[101X
  
  [33X[1;0Y[29X[2XIwasawaTripleWithSubgroup[102X( [3XG[103X, [3XX[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis  function  returns  an  Iwasawa triple for a [3Xp[103X-group [3XG[103X such that [3XX[103X is a
  member  of it, if such a triple exists, and [9Xfail[109X otherwise. This function is
  used as an auxiliary function to compute an Iwasawa triple for a group [3XG[103X.[133X
  
  [33X[0;0YAn  Iwasawa  triple  for  a  [22Xp[122X-group [22XG[122X is a triple [22X(X,b,s)[122X such that [22XX[122X is an
  abelian  normal  subgroup  of  [22XG[122X with cyclic quotient, [22Xb[122X is a generator of a
  supplement  to  [22XX[122X  in [22XG[122X, and [22Xb[122X induces a power automorphism in [22XX[122X of the form
  [22Xx->  x^1+p^s[122X.  A  theorem  of  Iwasawa states that a [22Xp[122X-group [22XG[122X has a modular
  subgroup  lattice  (or, equivalently, [22XG[122X has all subgroups permutable) if and
  only  if  [22XG[122X  is  a  direct  product  of a quaternion group of order [22X8[122X and an
  elementary abelian [22X2[122X-group or [22XG[122X has an Iwasawa triple [22X(X,b,s)[122X with [22Xs ≥ 2[122X.[133X
  
  [33X[0;0YThe  construction  of  the  Iwasawa  triple  takes a generator [22Xb[122X of a cyclic
  supplement  to  [22XX[122X  in [22XG[122X.  Then we consider a generator [22Xa[122X of [22XX[122X of the largest
  possible order and find an element [22Xc[122X of [22X⟨ b ⟩[122X and an element [22Xs[122X such that [22Xa^c
  = a^1+p^s[122X. If such an element does not exist, the function returns [9Xfail[109X. For
  this  element, it checks whether for all generators [22Xt[122X of [22XX[122X, the equality [22Xt^c
  =  t^1+p^s[122X  holds. If this holds, it returns the triple [22X(X, c, s)[122X; otherwise
  it returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xe:=ExtraspecialGroup(27,9);[127X[104X
    [4X[28X<pc group of size 27 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIwasawaTripleWithSubgroup(e,Subgroup(e,[e.1,e.3]),3);[127X[104X
    [4X[28X[ Group([ f1, f3 ]), f2, 1 ][128X[104X
  [4X[32X[104X
  
  [1X5.3-4 IwasawaTriple[101X
  
  [33X[1;0Y[29X[2XIwasawaTriple[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YThis function computes an Iwasawa triple for the [22Xp[122X-group [3XG[103X, if it exists. If
  [3XG[103X  is not Iwasawa, the function returns [9Xfail[109X. If [3XG[103X is a direct product of an
  elementary  abelian [22X2[122X-group and a quaternion group of order [22X8[122X, it returns an
  empty list. If [3XG[103X is Iwasawa, then the function returns an Iwasawa triple for
  [3XG[103X.  An  Iwasawa  triple  for  a  group [3XG[103X is a triple [22X(X, b, s)[122X where [22XX[122X is an
  abelian  normal subgroup of [22XG[122X such that [22XG/X[122X is cyclic, [22Xb[122X is a generator of a
  cyclic  supplement  to  [22XX[122X in [22XG[122X, and [22Xs[122X is an integer such that for all [22Xx ∈ X[122X,
  [22Xx^b  =  x^1+p^s[122X.  A theorem of Iwasawa states that a [22Xp[122X-group [22XG[122X has a modular
  subgroup  lattice  (or, equivalently, [22XG[122X has all subgroups permutable) if and
  only  if  [22XG[122X  is  a  direct  product  of  an elementary abelian [22X2[122X-group and a
  quaternion group of order [22X8[122X or [22XG[122X has an Iwasawa triple [22X(X,b,s)[122X with [22Xs ≥ 2[122X if
  [22Xp = 2[122X.[133X
  
  [33X[0;0YThe  method  followed to find an Iwasawa triple for non-abelian non-Dedekind
  groups  begins  with  the whole group [22XG[122X. If the group is abelian, it returns
  the  Iwasawa triple [22X(G,1,log_pexp(G))[122X. If it is not abelian, it constructs a
  list [22Xl[122X formed by [22XG[122X. For every element [22XN[122X of [22Xl[122X, it takes the maximal subgroups
  of  [22XN[122X  which  are  normal  in  [22XG[122X  and  give cyclic quotient. If any of these
  subgroups is a member of an Iwasawa triple, it is computed with the function
  [2XIwasawaTripleWithSubgroup[102X  ([14X5.3-3[114X)  and  the value is returned. If not, [22XN[122X is
  removed  from  the  [22Xl[122X  and these maximal subgroups of [22XN[122X are added to [22Xl[122X. This
  follows  until  an  Iwasawa  triple  is  found or the list [22Xl[122X is empty. Since
  normal  subgroups  with  cyclic  quotient  are contained in a unique maximal
  chain, no subgroup appears twice in this algorithm.[133X
  
  [33X[0;0YThe  algorithm  also  takes  into  account  the fact that a Iwasawa group of
  exponent [22X4[122X  must  be  abelian  or  a direct product of a quaternion group of
  order [22X8[122X and an elementary abelian [22X2[122X-group.[133X
  
  [33X[0;0YFor  the trivial group, it returns the triple composed by the trivial group,
  its identity element, and the prime [22X3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xe:=ExtraspecialGroup(27,3);[127X[104X
    [4X[28X<pc group of size 27 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIwasawaTriple(e);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xe:=ExtraspecialGroup(27,9);[127X[104X
    [4X[28X<pc group of size 27 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIwasawaTriple(e);[127X[104X
    [4X[28X[ Group([ f1, f3 ]), f2, 1 ][128X[104X
  [4X[32X[104X
  
  [1X5.3-5 IsIwasawaSylow[101X
  
  [33X[1;0Y[29X[2XIsIwasawaSylow[102X( [3XG[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis  function  returns [9Xtrue[109X if [3XG[103X has an Iwasawa (modular) Sylow [3Xp[103X-subgroup,
  and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YRecall  that  a  [22Xp[122X-group  [22XP[122X has a modular subgroup lattice, or is an Iwasawa
  group,  when  all  subgroups of [22XP[122X are permutable. It is clear that a [22Xp[122X-group
  has a modular subgroup lattice if and only if it is a T-group.[133X
  
  [33X[0;0YThe  implementation  of  this  function  begins  by  searching for a pair of
  subgroups that do not permute. In this case, the function returns [9Xfalse[109X. The
  maximum  number  of  pairs  to be checked here is controlled by the variable
  [2XPermutMaxTries[102X  ([14X3.2-1[114X), which is assigned to [22X10[122X by default. If no such pair
  is found, the algorithm looks for an Iwasawa triple for a Sylow [22Xp[122X-subgroup [22XP[122X
  of [22XG[122X.  If  there exists one such triple [22X(X,b,s)[122X with [22Xs ≥ 2[122X when [22Xp = 2[122X or the
  group is a direct product of a quaternion group of order [22X8[122X and an elementary
  abelian [22X2[122X-group, then it returns [9Xtrue[109X; else it returns [9Xfalse[109X.[133X
  
  [33X[0;0YThe  values  of  the attributes [2XIsPTGroup[102X ([14X4.2-2[114X) and [2XIsTGroup[102X ([14X4.2-1[114X) for [22XP[122X
  are set by the function.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xe:=ExtraspecialGroup(27,9);[127X[104X
    [4X[28X<pc group of size 27 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsIwasawaSylow(e,3);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
