Page 77 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.

mož Moravec: Some Topics in the Theory of Finite Groups 65

gap> G := AlternatingGroup( 4 );
Alt( [ 1 .. 4 ] )
gap> Order( G );
12

One can also locate A4 within the list of all normal subgroups of S4:

gap> G := SymmetricGroup( 4 );
Sym( [ 1 .. 4 ] )
gap> norm := NormalSubgroups( G );
[ Sym( [ 1 .. 4 ] ), Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]),

Group([ (1,4)(2,3), (1,3)(2,4) ]), Group(()) ]
gap> List( norm, StructureDescription );
[ "S4", "A4", "C2 x C2", "1" ]
gap> Q := G / norm[ 2 ];
Group([ f1 ])
gap> StructureDescription( Q );
"C2"

We can also construct the natural homomorphism S4 → S4/A4 as follows:

gap> G := SymmetricGroup( 4 );;
gap> norm:= NormalSubgroups( G );;
gap> N:=norm[ 2 ];
Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ])
gap> hom := NaturalHomomorphismByNormalSubgroup( G, N );
[ (1,2,3,4), (1,2) ] -> [ f1, f1 ]
gap> Kernel( hom ) = N;
true
gap> StructureDescription( Image( hom ) );
"C2"

Linear groups

Let F be a field. The set of all invertible n ×n matrices over F is a group under multiplica-
tion. It is called the general linear group of dimension n over F , and denoted by GL(n, F ).
By Galois’ theorem, the order of a finite field is alwasy a prime power, and if q is a prime

power, then there is, up to isomorphism, a unique field of order q . It is denoted by GF(q ).
The group GL(n, GF(q )) is also denoted as GL(n,q ).

The determinant map det : GL(n , F ) → F × is clearly a surjective homomorphism of
groups. Its kernel is denoted by SL(n, F ) and called the special linear group of dimension
n over F . Its elements are precisely all the matrices A ∈ GL(n, F ) with det A = 1.