Strong Shoda pairs
Computing strong Shoda pairs
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 () of a finite group G.

gap> StrongShodaPairs( SymmetricGroup(4) ); [ [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ]) ], [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3) ]) ], [ Group([ (1,2)(3,4), (1,3,2,4), (3,4) ]), Group([ (1,2)(3,4), (1,3,2,4) ]) ], [ Group([ (1,2)(3,4), (3,4), (1,3,2,4) ]), Group([ (1,2)(3,4), (3,4) ]) ], [ Group([ (1,4)(2,3), (1,3)(2,4), (2,4,3) ]), 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([ ]) ] ]
Properties related with Shoda pairs
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 () 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
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 () 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
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The input G should be a finite group.

Returns true if G is a strongly monomial () 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