‣ ExtremelyStrongShodaPairs( 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 extremely strong Shoda pairs (9.16) of a finite group G.
gap> ExtremelyStrongShodaPairs(DihedralGroup(32)); [ [ <pc group of size 32 with 5 generators>, <pc group of size 32 with 5 generators> ], [ <pc group of size 32 with 5 generators>, Group([ f1*f2*f3*f4*f5, f3, f4, f5 ]) ], [ <pc group of size 32 with 5 generators>, Group([ f2, f3, f4, f5 ]) ], [ <pc group of size 32 with 5 generators>, Group([ f1, f3, f4, f5 ]) ], [ Group([ f1*f2*f3*f4*f5, f3, f4, f5 ]), Group([ f1*f2*f4*f5, f4, f5 ]) ], [ Group([ f2, f3, f4, f5 ]), Group([ f5 ]) ], [ Group([ f2, f3, f4, f5 ]), Group([ ]) ] ] gap> ExtremelyStrongShodaPairs(SL(2,3)); [ [ SL(2,3), SL(2,3) ], [ SL(2,3), Group([ [ [ Z(3)^0, Z(3)^0 ], [ Z(3)^0, Z(3) ] ], [ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ], [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]) ], [ Group([ [ [ Z(3)^0, Z(3)^0 ], [ Z(3)^0, Z(3) ] ], [ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ], [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]), Group([ [ [ 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ] ], [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]) ] ] gap> ExtremelyStrongShodaPairs(SymmetricGroup(5)); [ [ Sym( [ 1 .. 5 ] ), Sym( [ 1 .. 5 ] ) ], [ Sym( [ 1 .. 5 ] ), Alt( [ 1 .. 5 ] ) ] ]
‣ 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> ssp:=StrongShodaPairs( SymmetricGroup(4) );; gap> Length(ssp); 5 gap> List(ssp,x->List(x,StructureDescription)); [ [ "S4", "S4" ], [ "S4", "A4" ], [ "A4", "C2 x C2" ], [ "D8", "C2 x C2" ], [ "D8", "C4" ] ] gap> ssp:=StrongShodaPairs( DihedralGroup(64) );; gap> Length(ssp); 8 gap> List(ssp,x->List(x,StructureDescription)); [ [ "D64", "D64" ], [ "D64", "D32" ], [ "D64", "C32" ], [ "D64", "D32" ], [ "D32", "D16" ], [ "C32", "C4" ], [ "C32", "C2" ], [ "C32", "1" ] ]
‣ IsExtremelyStrongShodaPair( G, K, H ) | ( operation ) |
The first argument should be a finite group G, the second one a normal sugroup K of G and the third one a subgroup of K.
Returns true if (K,H) is an extremely strong Shoda pair (9.16) of G, and false otherwise.
gap> G:=SymmetricGroup(4);; K:=Group( (1,3,2,4), (3,4) );; gap> H1:=Group( (2,4,3), (1,4)(2,3), (1,3)(2,4) );; gap> H2:=Group( (3,4), (1,2)(3,4) );; gap> IsExtremelyStrongShodaPair( G, G, H1 ); true gap> IsExtremelyStrongShodaPair( G, K, H2 ); false gap> IsExtremelyStrongShodaPair( G, G, H2 ); false gap> IsExtremelyStrongShodaPair( G, G, K ); false
‣ 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.
Note that every extremely strong Shoda pair is a strong Shoda pair, but the converse is not true.
gap> G:=SymmetricGroup(4);; K:=Group( (1,3,2,4), (3,4) );; gap> H1:=Group( (2,4,3), (1,4)(2,3), (1,3)(2,4) );; gap> H2:=Group( (3,4), (1,2)(3,4) );; gap> IsStrongShodaPair( G, G, H1 ); true gap> IsExtremelyStrongShodaPair( G, K, H2 ); false gap> IsStrongShodaPair( G, K, H2 ); true gap> IsStrongShodaPair( G, G, K ); false
‣ 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
‣ IsStronglyMonomial( G ) | ( operation ) |
The input G should be a finite group.
Returns true if G is a strongly monomial (9.17) 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
‣ IsNormallyMonomial( G ) | ( operation ) |
The input G should be a finite group.
Returns true if G is a finite normally monomial (9.18) group.
gap> D24:=DihedralGroup(24); <pc group of size 24 with 4 generators> gap> IsNormallyMonomial(D24); true gap> G:=SmallGroup(192,1023); <pc group of size 192 with 7 generators> gap> IsNormallyMonomial(G); true gap> G:=SmallGroup(1029,12); <pc group of size 1029 with 4 generators> gap> IsNormallyMonomial(G); false gap> IsStronglyMonomial(G); true gap> G:=SL(2,3); SL(2,3) gap> IsNormallyMonomial(G); false gap> IsStronglyMonomial(G); false
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