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

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

In recent years, many authors have considered totally and mutually permutable subgroups. Recall that two subgroups A and B of a group G are *totally permutable* if every subgroup of A permutes with every subgroup of B, and they are *mutually permutable* if every subgroup of A permutes with B and every subgroup of B permutes with A.

We have defined some "One" functions which give a pair of subgroups which do not permute and prove that two subgroups fail to have a certain property.

We have also defined some functions to work with totally and mutually f-permutable subgroups, where f is a subgroup embedding functor.

The functions of this chapter are defined in a preliminary state.

`‣ AreMutuallyPermutableSubgroups` ( [G, ]A, B ) | ( function ) |

This function returns `true`

if the subgroups A and B of G are mutually permutable subgroups, that is, every subgroup of A permutes with B and every subgroup of B permutes with A, and `false`

otherwise. The method used here checks only that A permutes with all cyclic subgroups of B and that B permutes with all cyclic subgroups of A.

The method with two arguments assume that A and B have a common supergroup.

`‣ OnePairShowingNotMutuallyPermutableSubgroups` ( [G, ]A, B ) | ( function ) |

This function returns a pair of the form [ `A`, `V` ] with `V` a subgroup of `B` or of the form [ `W`, `B` ] with `W` a subgroup of `A` in which both subgroups do not permute, or `fail`

if this pair does not exist because the product is mutually permutable.

`‣ AreTotallyPermutableSubgroups` ( [G, ]A, B ) | ( function ) |

This function returns `true`

if the subgroups A and B of G are totally permutable, that is, every subgroup of A permutes with every subgroup of B, and `false`

otherwise. The method used here checks only that every cyclic subgroup of A permutes with every cyclic subgroup of B.

The method with two arguments assume that A and B have a common supergroup.

`‣ OnePairShowingNotTotallyPermutableSubgroups` ( [G, ]A, B ) | ( function ) |

This function returns a pair of the form [ `V`, `W` ], with `V` a subgroup of `A` and `W` a subgroup of `B`, such that both subgroups do not permute, or `fail`

if this pair does not exist because the product is totally permutable.

gap> g:=SymmetricGroup(4); Sym( [ 1 .. 4 ] ) gap> a:=AlternatingGroup(4); Alt( [ 1 .. 4 ] ) gap> b:=Subgroup(g,[(1,2,3,4),(1,3)]); Group([ (1,2,3,4), (1,3) ]) gap> AreMutuallyPermutableSubgroups(g,a,b); true gap> AreTotallyPermutableSubgroups(g,a,b); false gap> OnePairShowingNotTotallyPermutableSubgroups(g,a,b); [ Group([ (2,3,4) ]), Group([ (1,2)(3,4) ]) ] gap> c:=Subgroup(g,[(1,2,3)]); Group([ (1,2,3) ]) gap> AreMutuallyPermutableSubgroups(g,a,c); false gap> OnePairShowingNotMutuallyPermutableSubgroups(g,a,c); [ Group([ (2,3,4) ]), Group([ (1,2,3) ]) ] gap> AreMutuallyPermutableSubgroups(a,c); false gap> g:=SymmetricGroup(3); Sym( [ 1 .. 3 ] ) gap> a:=AlternatingGroup(3); Alt( [ 1 .. 3 ] ) gap> b:=Subgroup(g,[(1,2)]); Group([ (1,2) ]) gap> AreTotallyPermutableSubgroups(g,a,b); true

`‣ AreMutuallyFPermutableSubgroups` ( [G, ]A, B, fA, fB ) | ( function ) |

This function returns `true`

if the subgroups `A` and `B` are mutually `f`-permutable, and `false`

otherwise. Here `A` and `B` are subgroups of `G` and `fA` and `fB` are, respectively, lists of subgroups of `A` and `B`, respectively.

In the version with four arguments, A and B are assumed to be subgroups of a common supergroup.

`‣ OnePairShowingNotMutuallyFPermutableSubgroups` ( [G, ]A, B, fA, fB ) | ( function ) |

This function returns a pair of the form [ `A`, `V` ] with `V` a subgroup in `fB` or `B` or of the form [ `W`, `B` ] with `W` a subgroup in `fA` or `A` in which both subgroups do not permute, or `fail`

if this pair does not exist. Here `A` and `B` are subgroups of `G` and `fA` and `fB` are lists of subgroups of `A` and `B`, respectively.

In the version with four arguments, `A` and `B` are assumed to be subgroups of a common supergroup.

`‣ AreTotallyFPermutableSubgroups` ( [G, ]A, B, fA, fB ) | ( function ) |

This function returns `true`

if the subgroup `A` permutes with all subgroups in the list `fB` and `B` permutes with all subgroups in the list `fA`, and `false`

otherwise. Here `A` and `B` are subgroups of `G`, `fA` is a list of subgroups of `A` and `fB` is a list of subgroups of `B`.

In the version with four arguments, `A` and `B` are assumed to be subgroups of a common supergroup.

`‣ OnePairShowingNotTotallyFPermutableSubgroups` ( [G, ]A, B, fA, fB ) | ( function ) |

This function returns a pair of the form [ `U`, `V` ] with `U` a subgroup in `fA` or `A` and `V` a subgroup in `fB` or `B` in which both subgroups do not permute, or `fail`

if this pair does not exist. Here `A` and `B` are subgroups of `G`, `fA` is a list of subgroups of `A` and `fB` is a list of subgroups of `B`.

In the version with two arguments, `A` and `B` are assumed to be subgroups of a common supergroup.

gap> g:=SymmetricGroup(4); Sym( [ 1 .. 4 ] ) gap> a:=AlternatingGroup(4); Alt( [ 1 .. 4 ] ) gap> b:=Subgroup(g,[(1,2,3,4),(1,3)]); Group([ (1,2,3,4), (1,3) ]) gap> AreTotallyFPermutableSubgroups(g,a,b, > MaximalSubgroups(a),MaximalSubgroups(b)); false gap> OnePairShowingNotTotallyFPermutableSubgroups(g,a,b, > MaximalSubgroups(a),MaximalSubgroups(b)); [ Group([ (1,2,3) ]), Group([ (2,4), (1,3)(2,4) ]) ] gap> AreTotallyFPermutableSubgroups(g,a,b,DerivedSeries(a),DerivedSeries(b)); true

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