  
  [1X1 [33X[0;0YIntroduction to the [5XPERMUT[105X[101X[1X Package[133X[101X
  
  [33X[0;0Y[13XAll  functions  defined  in  this  package  deal  only  with  finite groups.[113X
  Moreover,  some of the functions assume that the orders of all subgroups are
  easily  computable  and  that the decomposition of the order of a group as a
  product of prime numbers can be done in a reasonable time.[133X
  
  [33X[0;0YThe  package  [5XPERMUT[105X  contains  some functions to deal with permutability in
  finite  groups.  It  includes  functions  to  test  some  subgroup embedding
  properties   related   to   permutability,   like   permutability  or  Sylow
  permutability.  It  also  includes  some  functions to check whether a group
  belongs to the classes of T-groups, PT-groups, and PST-groups, which are the
  classes   of   groups   in   which   normality,   permutability,  and  Sylow
  permutability, respectively, are transitive. These properties and classes of
  groups  have  been  widely  studied  during the last years. Most of them are
  described in [BBERA10].[133X
  
  [33X[0;0YThe  algorithms  for T-groups, PT-groups, and PST-groups of this package use
  some  interesting  local  descriptions  of groups in these classes, that is,
  given  in  terms  of some information related to the primes [22Xp[122X dividing their
  order,   usually  by  looking  at  [22Xp[122X-subgroups  or  [22Xp[122X-chief  factors.  These
  characterisations show that the only difference between all three classes of
  groups   in  the  soluble  universe  corresponds  to  the  Sylow  structure.
  Nevertheless,  for  the sake of completeness, we also provide functions that
  use  directly  the  definition of these classes. In the case of T-groups and
  PST-groups,  as  well  as  for  soluble  PT-groups,  we  reduce  the test to
  subnormal  subgroups  of defect [22X2[122X (see [BBERR07], [BBERR09], and [BBBC+09]).
  Of  course, to do this we must introduce some functions to check whether two
  subgroups permute and whether a subgroup is permutable or S-permutable.[133X
  
  [33X[0;0YSome  of  the definitions of group-related concepts appear more than once in
  this  manual,  in  the  description  of  different functions. Although these
  repetitions may seem unnecessary when reading the whole manual, we hope that
  they benefit users who read the online help in [5XGAP[105X.[133X
  
  [33X[0;0YIn  order  to  obtain  easily  counterexamples  which show that a group or a
  subgroup  does  not  satisfy  a certain property, we have introduced what we
  have  called [21XOne[121X functions, which store such counterexamples. In some cases,
  the  property  can  be  checked by proving that these counterexamples do not
  exist.[133X
  
  [33X[0;0YThis package requires the [5XFormat[105X package by B. Eick and C. R. B. Wright (see
  [EW03]),  because  it  uses  the functions [2XPResidual[102X ([14XFORMAT: PResidual[114X) and
  [2XSystemNormalizer[102X  ([14XFORMAT:  SystemNormalizer[114X), which are defined there. Some
  of the examples in this manual use the library of groups of small order.[133X
  
  [33X[0;0YThe  mathematical  foundations  of  the algorithms presented in this package
  have been described in [BBCLER13].[133X
  
  [33X[0;0YThe  authors  acknowledge the support of the grants MTM2010-19938-C03-01 and
  MTM2014-54707-C3-1-P  funded by the [13XMinisterio de Economía y Competitividad[113X,
  Spanish   Government   (all   authors),   PGC2018-095140-B-I00,   funded  by
  MCIN/AEI/10.13039/501100011033  and  by ``ERDF A way of making Europe'' (all
  authors),  the  grant  PROMETEO/2017/057 funded by GVA/10.13039/501100003359
  (A.  Ballester-Bolinches  and  R.  Esteban-Romero),  the grant 11271085 from
  National  Natural  Science  Foundation of China (A. Ballester-Bolinches) and
  the  predoctoral grant AP2010-2764 from the [13XMinisterio de Educación[113X, Spanish
  Government  (E.  Cosme-Llópez). The authors are also indebted to the members
  of  the  [5XGAP[105X  council, especially Leonard Soicher, Alice Niemeyer, Max Horn,
  and  Alexander  Konovalov,  as  well as to the anonymous referees, for their
  comments which have helped us to improve the package and its documentation.[133X
  
