Page 78 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.

3.2 Basic notions and examples

Let us consider some examples using GAP:

gap> G := GL( 2, 4);
GL(2,4)
gap> Order( G );
180
gap> el := Elements( G );;
gap> a := el[ 5 ];
[ [ 0*Z(2), Z(2)^0 ], [ Z(2^2), 0*Z(2) ] ]
gap> b := el[ 7 ];
[ [ 0*Z(2), Z(2)^0 ], [ Z(2^2), Z(2^2) ] ]
gap> Determinant( a );
Z(2^2)
gap> a * b^2;
[ [ Z(2^2)^2, Z(2)^0 ], [ Z(2^2)^2, Z(2^2)^2 ] ]
gap> H := SL( 2, 4 );
SL(2,4)
gap> Order( H );
60
gap> StructureDescription( H );
"A5"

Proposition 3.2.9 | GL(n ,q )| = (q n − 1)(q n − q ) · · · (q n − q n−1).

PROOF. A matrix is invertible if and only if its rows are linearly independent. This holds if
and only if the first row is non-zero and, for k = 2, . . . , n, the k -th row is not in the sub-
space spanned by the first k −1 rows. The number of possible rows is q n , and the number
lying in any k -dimensional subspace is q k . So the number of choices for the first row is
q n − 1, and for k = 2, . . . , n , the number of choices for the k -th row is q n − q k −1. Multi-
plying these, we get the formula.

Corollary 3.2.10 | SL(n,q )| = | GL(n,q )|/(q − 1).

PROOF. Let F = GF(q ). We already saw above that GL(n ,q )/ SL(n ,q ) =∼ F ×, and this gives

the result.

Dihedral groups

A symmetry of a figure in Euclidian space is a rigid motion (or a combination of a rigid
motion with reflection) of the space that carries the figure to itself. If we think of a rigid
motion as a linear map of the real vector space, then it can be represented by a matrix.
Alternatively, if we label the vertices of the figure, then a symmetry can be represented as
a permutation of these labels.