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permut

A package to deal with permutability in finite groups

2.0.5

11 January 2024

Adolfo Ballester-Bolinches
Email: Adolfo.Ballester@uv.es
Address:
Departament de Matemàtiques
Universitat de València
Dr. Moliner, 50
46100 Burjassot, València, Spain

Enric Cosme-Llópez
Email: Enric.Cosme@uv.es
Homepage: https://www.uv.es/coslloen
Address:
Departament de Matemàtiques
Universitat de València
Dr. Moliner, 50
46100 Burjassot, València, Spain

Ramón Esteban-Romero
Email: Ramon.Esteban@uv.es
Homepage: https://www.uv.es/estebanr

Contents

1 Introduction to the PERMUT Package

2 Installation and Help of the PERMUT Package

3 Permutability of Subgroups in Finite Groups

3.1 Permutability functions

3.1-1 ArePermutableSubgroups

3.2 Embedding properties related to permutability

  3.2-1 PermutMaxTries
  3.2-2 IsPermutable
  3.2-3 OneSubgroupNotPermutingWith
  3.2-4 IsFPermutable
  3.2-5 OneFSubgroupNotPermutingWith
  3.2-6 IsSPermutable
  3.2-7 OneSylowSubgroupNotPermutingWith
  3.2-8 IsSNPermutable
  3.2-9 OneSystemNormaliserNotPermutingWith
  3.2-10 IsConjugatePermutable
  3.2-11 OneConjugateSubgroupNotPermutingWith
  3.2-12 IsWeaklySPermutable
  3.2-13 OneElementShowingNotWeaklySPermutable
  3.2-14 IsWeaklyPermutable
  3.2-15 OneElementShowingNotWeaklyPermutable
  3.2-16 IsWeaklyNormal
  3.2-17 OneElementShowingNotWeaklyNormal
  3.2-18 IsWithSubnormalizerCondition
  3.2-19 OneSubgroupInWhichSubnormalNotNormal
  3.2-20 IsWithSubpermutizerCondition
  3.2-21 OneSubgroupInWhichSubnormalNotPermutable
  3.2-22 IsWithSSubpermutizerCondition
  3.2-23 OneSubgroupInWhichSubnormalNotSPermutable

4 T-groups, PT-groups, and PST-groups

4.1 "One" functions

  4.1-1 OneSubnormalNonNormalSubgroup
  4.1-2 OneSubnormalNonPermutableSubgroup
  4.1-3 OneSubnormalNonSPermutableSubgroup
  4.1-4 OneSubnormalNonConjugatePermutableSubgroup
  4.1-5 OneSubnormalNonSNPermutableSubgroup

4.2 Group properties related to permutability

  4.2-1 IsTGroup
  4.2-2 IsPTGroup
  4.2-3 IsPSTGroup
  4.2-4 IsCPTGroup
  4.2-5 IsPSNTGroup

5 Local Functions in the PERMUT Package

5.1 A Local Function for Supersolubility

5.1-1 IsPSupersolvable

5.2 Local functions for T-groups, PT-groups, and PST-groups

  5.2-1 IsCp
  5.2-2 IsXp
  5.2-3 IsYp

5.3 Auxiliary Functions for T-groups, PT-groups, and PST-groups

  5.3-1 IsAbCp
  5.3-2 IsDedekindSylow
  5.3-3 IwasawaTripleWithSubgroup
  5.3-4 IwasawaTriple
  5.3-5 IsIwasawaSylow

6 Totally and Mutually Permutable Products

6.1 Functions for Mutually and Totally Permutable Products

  6.1-1 AreMutuallyPermutableSubgroups
  6.1-2 OnePairShowingNotMutuallyPermutableSubgroups
  6.1-3 AreTotallyPermutableSubgroups
  6.1-4 OnePairShowingNotTotallyPermutableSubgroups
  6.1-5 AreMutuallyFPermutableSubgroups
  6.1-6 OnePairShowingNotMutuallyFPermutableSubgroups
  6.1-7 AreTotallyFPermutableSubgroups
  6.1-8 OnePairShowingNotTotallyFPermutableSubgroups

7 Other Functions in the PERMUT Package

  7.1-1 AllSubnormalSubgroups
  7.1-2 PrimesDividingSize
  7.1-3 SylowSubgroups
  7.1-4 IsSCGroup
  7.1-5 IsSylowTowerGroup
  7.1-6 Permutizer
  7.1-7 AllGeneratorsCyclicPGroup

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