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3 Strong Shoda pairs

3.1 Computing strong Shoda pairs

3.1-1 StrongShodaPairs
 ‣ StrongShodaPairs( G ) ( attribute )

Returns: A list of pairs of subgroups of the input group.

The input should be a finite group G.

Computes a list of representatives of the equivalence classes of strong Shoda pairs (9.15) of a finite group G.


gap> StrongShodaPairs( SymmetricGroup(4) );
[ [ Sym( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ]) ],
[ Sym( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,3)(2,4), (2,4,3) ]) ],
[ Group([ (3,4), (1,3,2,4) ]), Group([ (1,3,2,4), (1,2)(3,4) ]) ],
[ Group([ (1,3,2,4), (3,4) ]), Group([ (3,4), (1,2)(3,4) ]) ],
[ Group([ (2,4,3), (1,4)(2,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]) ] ]
gap> StrongShodaPairs( DihedralGroup(64) );
[ [ <pc group of size 64 with 6 generators>,
Group([ f6, f5, f4, f3, f1, f2 ]) ],
[ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1*f2 ])
],
[ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f2 ]) ],
[ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1 ]) ],
[ Group([ f1*f2, f4*f5*f6, f5*f6, f6, f3, f3 ]),
Group([ f6, f5, f4, f1*f2 ]) ],
[ Group([ f6, f5, f2, f3, f4 ]), Group([ f6, f5 ]) ],
[ Group([ f6, f2, f3, f4, f5 ]), Group([ f6 ]) ],
[ Group([ f2, f3, f4, f5, f6 ]), Group([  ]) ] ]



3.2 Properties related with Shoda pairs

3.2-1 IsStrongShodaPair
 ‣ IsStrongShodaPair( G, K, H ) ( operation )

The first argument should be a finite group G, the second one a sugroup K of G and the third one a subgroup of K.

Returns true if (K,H) is a strong Shoda pair (9.15) of G, and false otherwise.


gap> G:=SymmetricGroup(3);; K:=Group([(1,2,3)]);; H:=Group( () );;
gap> IsStrongShodaPair( G, K, H );
true
gap> IsStrongShodaPair( G, G, H );
false
gap> IsStrongShodaPair( G, K, K );
false
gap> IsStrongShodaPair( G, G, K );
true



3.2-2 IsShodaPair
 ‣ IsShodaPair( G, K, H ) ( operation )

The first argument should be a finite group G, the second a subgroup K of G and the third one a subgroup of K.

Returns true if (K,H) is a Shoda pair (9.14) of G.

Note that every strong Shoda pair is a Shoda pair, but the converse is not true.


gap> G:=AlternatingGroup(5);;
gap> K:=AlternatingGroup(4);;
gap> H := Group( (1,2)(3,4), (1,3)(2,4) );;
gap> IsStrongShodaPair( G, K, H );
false
gap> IsShodaPair( G, K, H );
true



3.2-3 IsStronglyMonomial
 ‣ IsStronglyMonomial( G ) ( operation )

The input G should be a finite group.

Returns true if G is a strongly monomial (9.16) finite group.


gap> S4:=SymmetricGroup(4);;
gap> IsStronglyMonomial(S4);
true
gap> G:=SmallGroup(24,3);;
gap> IsStronglyMonomial(G);
false
gap> IsMonomial(G);
false
gap> G:=SmallGroup(1000,86);;
gap> IsMonomial(G);
true
gap> IsStronglyMonomial(G);
false


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