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### 5 Groups and homomorphisms

#### 5.1 Functions for groups

##### 5.1-1 Comm
 ‣ Comm( L ) ( operation )

This method has been transferred from package ResClasses.

It provides a method for Comm when the argument is a list (enclosed in square brackets), and calls the function LeftNormedComm.


gap> Comm( [ (1,2), (2,3) ] );
(1,2,3)
gap> Comm( [(1,2),(2,3),(3,4),(4,5),(5,6)] );
(1,5,6)
gap> Comm(Comm(Comm(Comm((1,2),(2,3)),(3,4)),(4,5)),(5,6));  ## the same
(1,5,6)



##### 5.1-2 IsCommuting
 ‣ IsCommuting( a, b ) ( operation )

This function has been transferred from package ResClasses.

It tests whether two elements in a group commute.


gap> D12 := DihedralGroup( 12 );
<pc group of size 12 with 3 generators>
gap> SetName( D12, "D12" );
gap> a := D12.1;;  b := D12.2;;
gap> IsCommuting( a, b );
false



##### 5.1-3 ListOfPowers
 ‣ ListOfPowers( g, exp ) ( operation )

This function has been transferred from package RCWA.

The operation ListOfPowers(g,exp) returns the list [g,g^2,...,g^exp] of powers of the element g.


gap> ListOfPowers( 2, 20 );
[ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384,
32768, 65536, 131072, 262144, 524288, 1048576 ]
gap> ListOfPowers( (1,2,3)(4,5), 12 );
[ (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), (),
(1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), () ]
gap> ListOfPowers( D12.2, 6 );
[ f2, f3, f2*f3, f3^2, f2*f3^2, <identity> of ... ]



##### 5.1-4 GeneratorsAndInverses
 ‣ GeneratorsAndInverses( G ) ( operation )

This function has been transferred from package RCWA.

This operation returns a list containing the generators of G followed by the inverses of these generators.


gap> GeneratorsAndInverses( D12 );
[ f1, f2, f3, f1, f2*f3^2, f3^2 ]
gap> GeneratorsAndInverses( SymmetricGroup(5) );
[ (1,2,3,4,5), (1,2), (1,5,4,3,2), (1,2) ]



##### 5.1-5 UpperFittingSeries
 ‣ UpperFittingSeries( G ) ( attribute )
 ‣ LowerFittingSeries( G ) ( attribute )
 ‣ FittingLength( G ) ( attribute )

These three functions have been transferred from package ResClasses.

The upper and lower Fitting series and the Fitting length of a solvable group are described here: https://en.wikipedia.org/wiki/Fitting_length.


gap> UpperFittingSeries( D12 );  LowerFittingSeries( D12 );
[ Group([  ]), Group([ f3, f2*f3 ]), Group([ f3, f2*f3, f1 ]) ]
[ D12, Group([ f3 ]), Group([  ]) ]
gap> FittingLength( D12 );
2
gap> S4 := SymmetricGroup( 4 );;
gap> UpperFittingSeries( S4 );
[ Group(()), Group([ (1,2)(3,4), (1,4)(2,3) ]), Group([ (1,2)(3,4), (1,4)
(2,3), (2,4,3) ]), Group([ (3,4), (2,3,4), (1,2)(3,4) ]) ]
gap> List( last, StructureDescription );
[ "1", "C2 x C2", "A4", "S4" ]
gap> LowerFittingSeries( S4 );
[ Sym( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,3)
(2,4) ]), Group(()) ]
gap> List( last, StructureDescription );
[ "S4", "A4", "C2 x C2", "1" ]
gap> FittingLength( S4);
3



#### 5.2 Functions for group homomorphisms

##### 5.2-1 EpimorphismByGenerators
 ‣ EpimorphismByGenerators( G, H ) ( operation )

This function has been transferred from package RCWA.

It constructs a group homomorphism which maps the generators of G to those of H. Its intended use is when G is a free group, and a warning is printed when this is not the case. Note that anything may happen if the resulting map is not a homomorphism!


gap> G := Group( (1,2,3), (3,4,5), (5,6,7), (7,8,9) );;
gap> phi := EpimorphismByGenerators( FreeGroup("a","b","c","d"), G );
[ a, b, c, d ] -> [ (1,2,3), (3,4,5), (5,6,7), (7,8,9) ]
gap> PreImagesRepresentative( phi, (1,2,3,4,5,6,7,8,9) );
d*c*b*a
gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;;
gap> d*c*b*a;
(1,2,3,4,5,6,7,8,9)
gap> ## note that it is easy to produce nonsense:
gap> epi := EpimorphismByGenerators( Group((1,2,3)), Group((8,9)) );
Warning: calling GroupHomomorphismByImagesNC without checks
[ (1,2,3) ] -> [ (8,9) ]
gap> IsGroupHomomorphism( epi );
true
gap> Image( epi, (1,2,3) );
()
gap> Image( epi, (1,3,2) );
(8,9)


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