GAP (Wedderga) - Chapter 3: Shoda pairs

Goto Chapter: Top 1 2 3 4 5 6 7 8 9 Bib Ind
[Top of Book] [Contents] [Previous Chapter] [Next Chapter]

[MathJax off]

3 Shoda pairs
3.1 Computing extremely strong Shoda pairs
3.1-1 ExtremelyStrongShodaPairs
3.2 Computing strong Shoda pairs
3.2-1 StrongShodaPairs
3.3 Properties related with Shoda pairs
3.3-1 IsExtremelyStrongShodaPair
3.3-2 IsStrongShodaPair
3.3-3 IsShodaPair
3.3-4 IsStronglyMonomial
3.3-5 IsNormallyMonomial

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

#### 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

[Top of Book] [Contents] [Previous Chapter] [Next Chapter]
Goto Chapter: Top 1 2 3 4 5 6 7 8 9 Bib Ind

generated by GAPDoc2HTML