GAP (Utils) - Chapter 4: Groups and homomorphisms

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4 Groups and homomorphisms
4.1 Functions for groups
4.1-1 Comm
4.1-2 IsCommuting
4.1-3 ListOfPowers
4.1-4 GeneratorsAndInverses
4.1-5 UpperFittingSeries
4.2 Functions for group homomorphisms
4.2-1 EpimorphismByGenerators

### 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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