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4 Groups and homomorphisms
 4.1 Functions for groups
 4.2 Functions for group homomorphisms

4 Groups and homomorphisms

4.1 Functions for groups

4.1-1 Comm
‣ Comm( L )( operation )

This method is in the process of being transferred from package ResClasses: for now you should LoadPackage("resclasses") in order to use it. 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)

4.1-2 IsCommuting
‣ IsCommuting( a, b )( operation )

This function is in the process of being transferred from package ResClasses: for now you should LoadPackage("resclasses") in order to use it. It tests whether two elements in a group commute.


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

4.1-3 ListOfPowers
‣ ListOfPowers( g, exp )( operation )

This function is in the process of being transferred from package RCWA: for now you should LoadPackage("rcwa") in order to use it.

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


gap> ListOfPowers( D12.2, 6 );
[ f2, f3, f2*f3, f3^2, f2*f3^2, <identity> of ... ]

4.1-4 GeneratorsAndInverses
‣ GeneratorsAndInverses( G )( operation )

This function is in the process of being transferred from package RCWA: for now you should LoadPackage("rcwa") in order to use it.

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 ]

4.1-5 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 );
[ Group([  ]), Group([ f3, f2*f3 ]), Group([ f3, f2*f3, f1 ]) ]
gap> LowerFittingSeries( D12 );
[ D12, Group([ f3 ]), Group([  ]) ]
gap> FittingLength( D12 );
2

4.2 Functions for group homomorphisms

4.2-1 EpimorphismByGenerators
‣ EpimorphismByGenerators( G, H )( attribute )

This function is in the process of being transferred from package RCWA: for now you should LoadPackage("rcwa") in order to use it. It maps the generators of \(G\) to those of \(H\). It is not checked that this map is a group homomorphism!


gap> G := Group((1,2,3,4),(3,4));;
gap> H := Group((6,7),(7,8));;    
gap> e1 := EpimorphismByGenerators(G,H);
[ (1,2,3,4), (3,4) ] -> [ (6,7), (7,8) ]
gap> ## note that this is just an abbreviation for: 
gap> e2 := GroupHomomorphismByImages( G, H, 
>              GeneratorsOfGroup(G), GeneratorsOfGroup(H) );;
gap> e1 = e2; 
true
gap> ## but the following is nonsense: 
gap> e0 := EpimorphismByGenerators( Group((1,2,3)), Group((8,9)) );
[ (1,2,3) ] -> [ (8,9) ]
gap> IsGroupHomomorphism(e0);
true

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