GAP (Wedderga) - Chapter 3: Strong Shoda pairs

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3 Strong Shoda pairs
3.1 Computing strong Shoda pairs
3.1-1 StrongShodaPairs
3.2 Properties related with Shoda pairs
3.2-1 IsStrongShodaPair
3.2-2 IsShodaPair
3.2-3 IsStronglyMonomial

### 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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