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### 3 Shoda pairs

#### 3.1 Computing extremely strong Shoda pairs

##### 3.1-1 ExtremelyStrongShodaPairs
 ‣ 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 ] ) ] ]



#### 3.2 Computing strong Shoda pairs

##### 3.2-1 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> 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" ] ]



#### 3.3 Properties related with Shoda pairs

##### 3.3-1 IsExtremelyStrongShodaPair
 ‣ 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



##### 3.3-2 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.

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



##### 3.3-3 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



##### 3.3-4 IsStronglyMonomial
 ‣ 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



##### 3.3-5 IsNormallyMonomial
 ‣ 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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