GAP (PERMUT) - Chapter 6: Totally and Mutually Permutable Products

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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 Totally and Mutually Permutable Products

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.

#### 6.1 Functions for Mutually and Totally Permutable Products

##### 6.1-1 AreMutuallyPermutableSubgroups

`‣ 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 assumes that A and B have a common supergroup.

##### 6.1-2 OnePairShowingNotMutuallyPermutableSubgroups

`‣ 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.

##### 6.1-3 AreTotallyPermutableSubgroups

`‣ 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 assumes that A and B have a common supergroup.

##### 6.1-4 OnePairShowingNotTotallyPermutableSubgroups

`‣ 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

##### 6.1-5 AreMutuallyFPermutableSubgroups

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

This function returns `true` if A permutes with B and all elements of fB and B permutes with all elements of fA, 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.

##### 6.1-6 OnePairShowingNotMutuallyFPermutableSubgroups

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

This function returns a pair of the form [ A, V ] with V equal to B or V a subgroup in fB, or of the form [ W, B ] with W equal to A or W a subgroup in fA, 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.

##### 6.1-7 AreTotallyFPermutableSubgroups

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

This function returns `true` if the subgroup A and all elements of the list fA permute with B and all subgroups in the list fB, 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.

##### 6.1-8 OnePairShowingNotTotallyFPermutableSubgroups

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

This function returns a pair of the form [ U, V ], with U equal to A or U a subgroup in fA, and V equal to B or V a subgroup in fB, 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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