Part 19: Linear groups over other fields

Vector spaces over fields

 You have met fields other than the reals and rationals before, for example the complex numbers C and the p--element fields Z(p). There are many others, such as the field of fractions of polynomials in n variables with real coefficients. It can be shown also that for every prime power q = pn there is a field with q elements, unique up to isomorphism. An important fact about finite fields of order q is that the multiplicative group of non-zero elements is cyclic of order q-1.

 We are interested mainly in finite dimensional vector spaces over fields and the group of non--singular linear transformations of such vector spaces. They are defined in a similar manner to vector spaces over the reals, and they have matrices and determinants calculated in the same way.

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General and Special Linear Groups

 We shall denote an arbitrary field by F, vector spaces V will be F-spaces and GL(n,F) is the group of non-singular linear transformations of V or equivalently the group of invertible nxn matrices over F if V has dimension n.

 Example 1. The order of GL(n,F) when F has order q is just the number of nxn invertible matrices over F. There are qn - 1 possibilities for row 1, and once row 1 is chosen there are q vectors linearly dependent on it, so qn-q possibilities for row 2. Continuing in this way, there are qn-qr possibilities for row r+1. 

Hence |GL(n,F)| = (qn-1)(qn-q)..(qn-qn-1) = qn(n-1)/2(qn-1)(qn-1-1)..(q-1).

 For example, |GL(2,Z(2))| = 6 and |GL(2,Z(3))| = 48.

Return now to the case where F is an arbitrary field. Denote the multiplicative group of non-zero elements of F by F*. The map G -> det G: GL(n,F) -> F* is an epimorphism with kernel SL(n,F), the special linear group of degree n over F, consisting of the invertible matrices of determinant 1. If F* has q elements then F* has q-1, so in that case |SL(n,F)| = |GL(n,F)|/(q-1). In particular this implies that GL(n,Z(2)) = GL(n,Z(2)).

 

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Affine groups

 We define the affine group of degree n over a field F to be the set 

Aff(n,F) = {TL,a: L in GL(n,F) and a in Fn}, with multiplication TL,a.TM,b = TLM,aM+b. Of course this agrees with Aff(n) if F = R.

 Aff(n,F) is a group with identity TI,0 and (TL,a)-1 = TL-1,-aL-1.

 Aff(n,F) acts on the vector space Fn by bTL,a = bL+a. This action is faithful and Aff(n,F) is a subgroup of the group of permutations of Aff(n,F)n containing GL(n,F).

 If F is finite of order q, then obviously |Aff(n,F)| = qn|GL(n,F)| and in some small examples, this enables us to determine Aff(n,F).

 For example, Aff(1,Z(2)) is the group of functions {x-> x, x-> x+1} of order 2;

 Aff(1, Z(3)) has order 6 and is non-abelian so is isomorphic to S3.

 Aff(2, Z(2)): there are 6 elements in GL(2, Z(2)) and 4 in Z(2)2, so Aff(2, Z(2)) is a subgroup of S4 of order 24, so is isomorphic to S4.

 There is no direct analogue of the orthogonal groups, because there is none of Pythagoras Theorem. There are various equivalent constructions using non-degenerate inner products, but we don't deal with them in this course. 

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Projective Groups

 GL(n,F) also has a subgroup isomorphic to F*, namely the group of scalar matrices Z ={aI: a in F*} which is just the centre of GL(n,F) and hence is normal. The factor group GL(n,F)/Z is called the projective linear group over F of degree n, PGL(n,F). So in case F has order q, |PGL(n,F)| = |SL(n,F)| although of course they are not in general isomorphic as groups.

 Finally note that the centre of SL(n,F) is Z 0 SL(n,F) = 
{aI: an = 1}. It is normal in SL(n,F) and the factor group is the projective special linear group PSL(n,F) = (SL(n,F).Z)/Z. In case F has order q, 

|SL(n,F) 0 Z| = d = g.c.d.(n, q-1) from which we compute |PSL(n,F)| = |GL(n,F)|/ d.q-1. 

For example, |PSL(2,Z(2)| = 6 and |PSL(2,Z(3)| = 12. 

By Sylow's Theorems, neither of these groups is simple. But a major theorem of Group Theory, which we do not prove in this course is that PSL(n,F) is simple for n > 2 and for all fields F.

 Projective geometry Let F be a field and n a positive integer. Define Sk to be the set of k+1-dimensional subspaces of Fn. Since GL(n,F) preserves dimensions of subspaces, GL(n,F) acts on Sk. The kernel of this action is just Z ={aI: a in F*}, so in fact PGL(n,F) is a group acting on Sk.

 We can put a geometrical structure on Fn by defining points to be the elements of S0, lines to be elements of S1, planes to be elements of S2 and so on. In this structure, we see that any two lines intersect in a unique point, any two planes intersect in a unique line, and so on. Also, each pair of points is contained in a unique line, each pair of lines is contained in a unique plane etc. The whole structure is called projective geometry over F,and its natural group of transformations is PGL(n,F).

 The Moebius Group The linear fractional group LF(F) over F is the group 

{fabcd: F -> F, x -> (ax+b)/(cx+d), where ad-bc not= 0}

Theorem 22.1 LF(F) is a group of functions isomorphic to PSL(2,F).

Proof. First define Q: GL(2,F) -> LF(F) by

a b -> fabcd
c d

It is routine to check that Q is a homomorphism which is obviously onto.

 It is also routine to check that 

a b
c d
maps to the identity function if and only if a = d and c = b = 0. That is the matrix is aI2 for some non-zero a in F. 

By the fundamental homomorphism theorem for groups, LF(F) is isomorphic to PSL(2,F).

 Note that if for some scalar s, a'=sa, b'=sb, c'=sc and d'=sd, then fabcd = fa'b'c'd'.

 If F = C, the complex field, LF(C) is called the Moebius group. It is important in complex analysis. In this case, (and more generally whenever F is a field in which every element has a square root) given 

A =  a b
c d
with ad - bc not= 0, let s = 1/(square root(det A)), and let a'=sa, b'=sb, c'=sc and d'=sd. Then fabcd = fa'b'c'd' with a'd' - b'c' = 1, so 
A' =  a' b'
c' d'
is in SL(2, C).

 In other words, LF(C) = PGL(2,C) = PSL(2,C) = SL(2,C)/{+1,-1}. 

 

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 Last update Jan 11, 2000

Author: Phill Schultz, schultz@maths.uwa.edu.au

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