Groups and homomorphisms
Functions for groups 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)
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
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 ... ]
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) ]
These three functions have been transferred from package &ResClasses;.

Fitting series 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
Functions for group homomorphisms 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)