1.4 Examples: The Linear Groups

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Let F be a field and let V be a finite-dimensional vector space over the

field F. Denote by GL(V ) the set of non-singular linear transformations

T : V ! V . Clearly GL(V ) is a group with respect to composition; call

this group the general linear group of the vector space V . If dim V = n,

and if we denote by GLn(F) the multiplicative group of invertible n by n

matrices over F, then choice of an ordered basis A = (v1, v2, . . . , vn) yields

an isomorphism

GL(V ) _= −! GLn(F), T 7! [T]A,

where [T]A is the matrix representation of T relative to the ordered basis

A.

An easy calculation reveals that the center of the general linear group

GL(V ) consists of the scalar transformations:

Z(GL(V )) = {_ · I| _ 2 F} _= F×,

where F× is the multiplicative group of nonzero elements of the field F.

Another normal subgroup of GL(V ) is the special linear group :

SL(V ) = {T 2 GL(V )| det T = 1}.

Finally, the projective linear group and projective special linear group are

defined respectively by setting

PGL(V ) = GL(V )/Z(GL(V )), PSL(V ) = SL(V )/Z(SL(V )).

If F = Fq is the finite field4 of q elements, it is customary to use the notations

GLn(q) = GLn(Fq), SLn(q) = SLn(Fq), PGLn(q) = PGLn(Fq), PSLn(q) =

PSLn(Fq). These are finite groups, whose orders are given by the following:

Proposition 1.4.1 The orders of the finite linear groups are given by

|GLn(q)| = qn(n−1)/2(qn − 1)(qn−1 − 1) · · · (q − 1).

|SLn(q)| = 1

q−1 |GLn(q)|.

|PGLn(q)| = |SLn(q)| = 1

q−1 |GLn(q)|.

|PSLn(q)| = 1

(n,q−1) |SLn(q)|.

4We discuss finite fields in much more detail in Section 2.4.

1.4. EXAMPLES: THE LINEAR GROUPS 15

Notice that the general and special linear groups GL(V ) and SL(V )

obviously act on the set of vectors in the vector space V . If we denote

V ] = V − {0}, then GL(V ) and SL(V ) both act transitively on V ], except

when dim V = 1 (see Exercise 1, below).

Next, set P(V ) = {one-dimensional subspaces of V }, the projective space

of V ; note that GL(V ), SL(V ), PGL(V ), and PSL(V ) all act on P(V ).

These actions turn out to be doubly transitive (Exercise 2).

A flag in the n-dimensional vector space V is a sequence of subspaces

Vi1 _ Vi2 _ · · · _ Vir _ V,

where dim Vij = ij , j = 1, 2, · · · , r. We call the flag [Vi1 _ Vi2 _ · · · _ Vir ]

a flag of type (i1 < i2 < · · · < ir). Denote by (i1 < i2 < · · · < ir) the set

of flags of type (i1 < i2 < · · · < ir).

Theorem 1.4.2 The groups GL(V ), SL(V ), PGL(V ) and PSL(V ) all act

transitively on (i1 < i2 < · · · < ir).

Exercises 1.4

1. Prove if dim V > 1, GL(V ) and SL(V ) act transitively on V ] = V −

{0}. What happens if dim V = 1?

2. Show that all of the groups GL(V ), SL(V ), PGL(V ), and PSL(V ) act

doubly transitively on the projective space P(V ).

3. Let V have dimension n over the field F, and consider the set (1 <

2 < · · · < n − 1) of complete flags . Fix a complete flag

F = [V1 _ V2 _ · · · _ Vn−1] 2 (1 < 2 < · · · < n − 1).

If G = GL(V ) and if B = StabG(F), show that B is isomorphic with

the group of upper triangular n×n invertible matrices over F. If F = Fq

is finite of order q = pk, where p is prime, show that B = NG(P) for

some p-Sylow subgroup P _ G.

4. The group SL2(Z) consisting of 2 × 2 matrices having integer entries

and determinant 1 is obviously a group (why?). Likewise, for any

positive integer n, SL2(Z/(n)) makes perfectly good sense and is a

group. Indeed, if we reduce matrices in SL2(Z) modulo n, then we

16 CHAPTER 1. GROUP THEORY

get a homomorphism _n : SL2(Z) ! SL2(Z/(n)). Prove that this

homomorphism is surjective. In particular, conclude that the group

SL2(Z) is infinite.

5. We set PSL2(Z/(n)) = SLn(Z/(n))/Z(SLn(Z/(n)); show that

|PSL2(Z/(n))| =

(

6 if n = 2,

n3

2

Q

p|n(1 − 1

p2 ) if n > 2,

where p ranges over the distinct prime factors of n.

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