General linear group of degree two

From Groupprops
Revision as of 20:58, 30 August 2009 by Vipul (talk | contribs) (Created page with '==Definition== The '''general linear group of degree two''' over a field <math>k</math> (respectively, over a unital ring <math>R</math>), is defined as the group, under mul…')
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Definition

The general linear group of degree two over a field k (respectively, over a unital ring R), is defined as the group, under multiplication, of invertible 2 \times 2 matrices with entries in k. It is denoted GL(2,k) (respectively, GL(2,R)).

For a prime power q, GL(2,q) or GL_2(q) denotes the general linear group of degree two over the field (unique up to isomorphism) with q elements.

Arithmetic functions

Here, q denotes the order of the finite field and the group we work with is GL(2,q).

Function Value Explanation
order q^3 - q = q(q-1)(q+1) q^2 - 1 options for first row, q^2 - q options for second row.
exponent q^3 - q = q(q-1)(q+1) There is an element of order q^2 - 1 and an element of order q.
number of conjugacy classes q^2 - 1 There are q(q-1) conjugacy classes of semisimple matrices and q - 1 conjugacy classes of unipotent matrices.

Group properties

Property Satisfied Explanation
Abelian group No The matrices \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix} and \begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix} don't commute.
Nilpotent group No PSL(2,q) is simple for q \ge 4, and we can check the cases q = 2, 3 separately.
Solvable group Yes if q = 2,3, no otherwise. PSL(2,q) is simple for q \ge 4.
Supersolvable group Yes if q - 2, no otherwise. PSL(2,q) is simple for q \ge 4, and we can check the cases q = 2, 3 separately.