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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

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.

6.1-2 OnePairShowingNotMutuallyPermutableSubgroups

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

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.

6.1-4 OnePairShowingNotTotallyPermutableSubgroups

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

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.

6.1-6 OnePairShowingNotMutuallyFPermutableSubgroups

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.

6.1-7 AreTotallyFPermutableSubgroups

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.

6.1-8 OnePairShowingNotTotallyFPermutableSubgroups

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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