67

Let V = F 2 , and let W be the vector space of all quadratic forms on V (not nec-

essarily non-degenerate). Then rk(W ) = 3; a typical element of W is the quadratic

form ux2 + vxy + wy2 , where we have represented a typical vector in V as (x, y).

We use the triple (u, v, w) of coef¬cients to represent this vector of w. Now GL(V )

acts on W by substitution on the variables in the quadratic form. In other words,

to the matrix

ab

A= ∈ GL(2, F)

cd

corresponds the map

ux2 + vxy + wy2 ’ u(ax + cy)2 + v(ax + cy)(bx + dy) + w(bx + dy)2

= (ua2 + vab + wb2 )x2 + (2uac + v(ad + bc) + 2wbd)xy

+(uc2 + vcd + wd 2 )y2 ,

which is represented by the matrix

«2

2

a 2ac c

ρ(A) = ab ad + bc cd ∈ GL(3, F).

b2 d2

2bd

We observe several things about this representation ρ of GL(2, F):

(a) The kernel of the representation is {±I}.

(b) det(ρ(A)) = (det(A))3 .

(c) The quadratic form Q(u, v, w) = 4uw ’ v2 is multiplied by a factor det(A)2

by the action of ρ(A).

Hence we ¬nd a subgroup of O(Q) which is isomorphic to SL± (2, F)/{±I},

where SL± (2, F) is the group of matrices with determinant ±1. Moreover, its

intersection with SL(3, F) is SL(2, F)/{±1}. In fact, these are the full groups

O(Q) and SO(Q) respectively.

We see that in this case,

P„¦(Q) ∼ PSL(2, F) if |F| > 2,

=

and this group is simple if |F| > 3.

68

Case n = 4, r = 2. Our strategy is similar. We take the rank 4 vector space

over F to be the space M 2—2 (F), the space of 2 — 2 matrices over F (where F is

any ¬eld). The determinant function on V is a quadratic form: Q(X) = det(X).

Clearly X is the sum of two hyperbolic planes (for example, the diagonal and the

antidiagonal matrices).

There is an action of the group GL(2, F) — GL(2, F) on X, by the rule

ρ(A, B) : X ’ A’1 XB.

We see that ρ(A, B) preserves Q if and only if det(A) = det(B), and ρ(A, B) is the

identity if and only if A = B = »I for some scalar ». So we have a subgroup of

O(Q) with the structure

((SL(2, F) — SL(2, F)) · F — )/{(»I, »I) : » ∈ F — }.

Moreover, the map T : X ’ X also preserves Q. It can be shown that together

these elements generate O(Q).

Exercise 6.3 Show that the above map T has determinant ’1 on V , while ρ(A, B)

has determinant equal to det(A)’2 det(B)2 . Deduce (from the information given)

that SO(Q) has index 2 in O(Q) if and only if the characteristic of F is not 2.

Exercise 6.4 Show that, in the above case, we have

P„¦(Q) ∼ PSL(2, F) — PSL(2, F)

=

if |F| > 3.

Exercise 6.5 Use the order formulae for ¬nite orthogonal groups to prove that

the groups constructed on vector spaces of ranks 3 and 4 are the full orthogonal

groups, as claimed.

6.2 Characteristic 2, odd rank

In the case where the bilinear form is degenerate, we show that the orthogonal

group is isomorphic to a symplectic group.

Theorem 6.1 Let F be a perfect ¬eld of characteristic 2. Let Q be a non-degenerate

quadratic form in n variables over F, where n is odd. Then O(Q) ∼ Sp(n ’ 1, F).

=

69

Proof We know that the bilinear form B is alternating and has a rank 1 radical,

spanned by a vector z, say. By multiplying Q by a scalar if necessary, we may

assume that Q(z) = 1. Let G be the group induced on V /Z, where Z = z . Then

G preserves the symplectic form.

The kernel K of the homomorphism from G to G ¬xes each coset of Z. Since

Q(v + az) = Q(v) + a2 ,

and the map a ’ a2 is a bijection of F, each coset of Z contains one vector with

each possible value of Q. Thus K = 1, and G ∼ G.

=

Conversely, let g be any linear transformation of V /Z which preserves the

symplectic form induced by B. The above argument shows that there is a unique

permutation g of V lifting the action of g and preserving Q. Note that, since g

induces g on V /Z, it preserves B. We claim that g is linear. First, take any two

vectors v, w. Then

Q(vg + wg) = Q(vg) + Q(wg) + B(vg, wg)

= Q(v) + Q(w) + B(v, w)

= Q(v + w)

= Q((v + w)g);

and the linearity of g shows that vg + wg and (v + w)g belong to the same coset

of Z, and so they are equal. A similar argument applies for scalar multiplication.

So G = Sp(n ’ 1, F), and the result is proved.

We conclude that, with the hypotheses of the theorem, O(Q) is simple except

for n = 3 or n = 5, F = GF(2). Hence O(Q) coincides with P„¦(Q) with these

exceptions.

We conclude by constructing some more 2-transitive groups. Let F be a per-

fect ¬eld of characteristic 2, and B a symplectic form on F 2m . Then the set Q (B)

of all quadratic forms which polarise to B is a coset of the set of “square-seminilear

maps” on V , those satisfying

L(x + y) = L(x) + L(y),

L(cx) = c2 L(x)

(these maps are just the quadratic forms which polarise to the zero bilinear form).

In the ¬nite case, where F = GF(q) (q even), there are thus q2m such quadratic

forms, and they fall into two orbits under Sp(2m, q), corresponding to the two

70

types of forms. The stabiliser of a form Q is the corresponding orthogonal group

O(Q). The number of forms of each type is the index of the corresponding orthog-

onal group in the symplectic group, which can be calculated to be qm (qm + µ)/2

for a form of type µ.

Now specialise further to F = GF(2). In this case, “square-semilinear” maps

are linear. So, given a quadratic form Q polarising to B, we have

Q (B) = {Q + L : L ∈ V — }.

Further, each linear form can be written as x ’ B(x, a) for some ¬xed a ∈ V . Thus,

there is an O(Q)-invariant bijection between Q (B) and V . By Witt™s Lemma,

O(Q) has just three orbits on V , namely

{0}, {x ∈ V : Q(x) = 0, x = 0}, {x ∈ V : Q(x) = 1}.

So O(Q) has just three orbits on Q (B), namely

Q µ (B) \ {Q}, Q ’µ (B),

{Q},

where Q has type µ and Q µ (B) is the set of all forms of type µ in Q (B).

It follows that Sp(2m, 2) acts 2-transitively on each of the two sets Q µ (B), with

cardinalities 2m’1 (2m + µ). The point stabiliser in these actions are Oµ (2m, 2).

Exercise 6.6 What isomorphisms between symmetric and classical groups are il-

lustrated by the above 2-transitive actions of Sp(4, 2)?

6.3 Transvections and root elements

We ¬rst investigate orthogonal transvections, those which preserve the non-degenerate

quadratic form Q on the F-vector space V .

Proposition 6.2 There are no orthogonal transvections over a ¬eld F whose char-

acteristic is different from 2. If F has characteristic 2, then an orthogonal transvec-

tion for a quadratic form Q has the form

x ’ x ’ Q(a)’1 B(x, a)a,

where Q(a) = 0 and B is obtained by polarising Q.

71

Proof Suppose that the transvection x ’ x + (x f )a preserves the quadratic form

Q, and let B be the associated bilinear form. Then

Q(x + (x f )a) = Q(x)

for all x ∈ V , whence

(x f )2 Q(a) + (x f )B(x, a) = 0.

If x f = 0, we conclude that (x f )Q(a) + B(x, a) = 0. Since this linear equation

holds on the complement of a hyperplane, it holds everywhere; that is, B(x, a) =

’(x f )Q(a) for all x.

If the characteristic is not 2, then B(a, a) = 2Q(a). Substituting x = a in the

above equation, using a f = 0, we see that B(a, a) = 0, so Q(a) = 0. But then

B(x, a) = 0 for all x, contradicting the nondegeneracy of B in this case.

So we may assume that the characteristic is 2. If B(x, a) = 0 for all x ∈ V , then

Q(a) = 0 by non-degeneracy. Otherwise, choosing x with B(x, a) = 0, we see that

again Q(a) = 0. Then

x f = ’Q(a)’1 B(x, a),

and the proof is complete. (Incidentally, the fact that f is non-zero now shows that

a is not in the radical of B.)

Exercise 6.7 In the characteristic 2 case, replacing a by »a for a = 0 does not

change the orthogonal transvection.

The fact that, if Q is a non-degenerate quadratic form in three variables with

Witt rank 1 shows that we can ¬nd analogues of transvections acting on three-

dimensional sections of V . These are called root elements, and they will be used

in our simplicity proofs.

A root element is a transformation of the form

x ’ x + aB(x, v)u ’ aB(x, u)v ’ a2 Q(v)B(x, u)u

where Q(u) = B(u, v) = 0. The group of all such transformations for ¬xed u, v sat-

isfying the above conditions, together with the identity, is called a root subgroup

Xu,v .

Exercise 6.8 Prove that the root elements are isometries of Q, that they have

determinant 1, and that the root subgroups are abelian. Show further that, if

Q(u) = 0, then the group

Xu = Xu,v : v ∈ u⊥

is abelian, and is isomorphic to the additive group of u⊥ / u .

72

Exercise 6.9 Write down the root subgroup Xu,v for the quadratic form Q(x1 , x2 , x3 ) =

2

x1 x3 ’ x2 relative to to the given basis {e1 , e2 , e3 }, where u = e1 and v = e2 .

Now the details needed to apply Iwasawa™s Lemma are similar to, but more

complicated than, those that we have seen in the cases of the other classical

groups. We summarise the important steps. Let Q be a quadratic form with Witt

rank at least 2, and not of Witt rank 2 on a vector space of rank 4 (that is, not

equivalent to x1 x2 + x3 x4 ). We also exclude the case where Q has Witt index 2 on

a rank 5 vector space over GF(2): in this case P„¦(Q) ∼ PSp(4, 2) ∼ S6 .

= =

(a) The root subgroups are contained in „¦(Q), the derived group of O(Q).

(b) The abelian group Xu is normal in the stabiliser of u.

(c) „¦(Q) is generated by the root subgroups.

(d) „¦(Q) acts primitively on the set of ¬‚at 1-spaces.

Note that the exception of the case of rank 4 and Witt index 2 is really necessary

for (d): the group „¦(Q) ¬xes the two families of rulings on the hyperbolic quadric

shown in Figure 1 on p. 41, and each family is a system of blocks of imprimitivity

for this group.

Then from Iwasawa™s Lemma we conclude:

Theorem 6.3 Let Q be a non-degenerate quadratic form with Witt rank at least 2,

but not of Witt rank 2 on either a vector space of rank 4 or a vector space of rank 5

over GF(2). Then P„¦(Q) is simple.

It remains for us to discover the order of P„¦(Q) over a ¬nite ¬eld. We give

the result here, and defer the proof until later. The facts are as follows.

Proposition 6.4 (a) Let Q have Witt index at least 2, and let F have character-

istic different from 2. Then SO(Q)/„¦(Q) ∼ F — /(F — )2 .

=

(b) Let F be a perfect ¬eld of characteristic 2 and let Q have Witt index at

least 2; exclude the case of a rank 4 vector space over GF(2). Then SO(Q) :

„¦(Q)| = 2.

73

The proof of part (a) involves de¬ning a homomorphism from SO(Q) to F — /(F — )2

called the spinor norm, and showing that it is onto and its kernel is „¦(Q) except

in the excluded case.

In the remaining cases, we work over the ¬nite ¬eld GF(q), and write O(n, q),

understanding that if n is even then Oµ (n, q) is meant.

Proposition 6.5 Excluding the case q even and n odd:

(a) | SO(n, q) : „¦(n, q)| = 2.

(b) For q odd, ’I ∈ „¦µ (2m, q) if and only if qm ∼ µ (mod 4).

=

The last part is proved by calculating the spinor norm of ’I. Putting this to-

gether with the order formula for SO(n, q) already noted, we obtain the following

result:

Theorem 6.6 For m ≥ 2, excluding the case P„¦+ (4, 2), we have

m’1

(q ’ µ) ∏ (q2i ’ 1)

µ

(4, qm ’ µ),

m(m’1) m

| P„¦ (2m, q)| = q

i=1

m

∏(q2i ’ 1)

m2

| P„¦(2m + 1, q)| = (2, q ’ 1).

q

i=1

Proof For q odd, have already shown that the order of SO(n, q) is given by the

expression in parentheses. We divide by 2 on passing to „¦(n, q), and another 2 on

factoring out the scalars if and only if 4 divides qm ’ µ. For q even, | SO(n, q)| is

twice the bracketed expression, and we lose the factor 2 on passing to „¦(n, q) =

P„¦(n, q).

Now we note that | P„¦(2m + 1, q)| = | PSp(2m, q)| for all m. In the case m = 1,

these groups are isomorphic, since they are both isomorphic to PSL(2, q). We have

also seen that they are isomorphic if q is even. We will see later that they are also

isomorphic if m = 2. However, they are non-isomorphic for m ≥ 3 and q odd.

This follows from the result of the following exercise.

Exercise 6.10 Let q be odd and m ≥ 2.

(a) The group PSp(2m, q) has m/2 + 1 conjugacy classes of elements of or-

der 2.

74

(b) The group P„¦(2m + 1, q) has m conjugacy classes of elements of order 2.

Hint: if t ∈ Sp(2m, q) or t ∈ „¦(2m + 1, q) = P„¦(2m + 1, q) satis¬es t 2 = 1, then

V = V + • V ’ , where vt = »v for v ∈ V » ; and the subspaces V + and V ’ are or-

thogonal. Show that there are m possibilities for the subspaces V + and V ’ up to

isometry; in the symplectic case, replacing t by ’t interchanges these two spaces

but gives the same element of PSp(2m, q). In the case PSp(2m, q), there is an

additional conjugacy class arising from elements t ∈ Sp(2m, q) with t 2 = ’1.

It follows from the Classi¬cation of Finite Simple Groups that there are at

most two non-isomorphic simple groups of any given order, and the only instances

where there are two non-isomorphic groups are

PSp(2m, q) ∼ P„¦(2m + 1, q) for m ≥ 3, q odd

=

and

PSL(3, 4) ∼ PSL(4, 2) ∼ A8 .

= =

The lecture course will not contain detailed proofs of the simplicity of P„¦(n, q),

but at least it is possible to see why PSO+ (2m, q) contains a subgroup of index 2

for q even. Recall from Chapter 3 that, for the quadratic form

x1 x2 + · · · + x2m’1 x2m

of Witt index m in 2m variables, the ¬‚at m-spaces fall into two families F + and

F ’ , with the property that the intersection of two ¬‚at m-spaces has even codimen-

sion in each if they belong to the same family, and odd codimension otherwise.

Any element of the orthogonal group must ¬x or interchange the two families.

Now, for q even, SO+ (2m, q) contains an element which interchanges the two

families: for example, the transformation which interchanges the coordinates x1

and x2 and ¬xes all the others. So SO+ (2m, q) has a subgroup of index 2 ¬x-

ing the two families, which is „¦+ (2m, q). (In the case where q is odd, such a

transformation has determinant ’1.)

75

7 Klein correspondence and triality

The orthogonal groups in dimension up to 8 have some remarkable properties.

These include, in the ¬nite case,

(a) isomorphisms between classical groups:

“ P„¦’ (4, q) ∼ PSL(2, q2 ),

=

“ P„¦(5, q) ∼ PSp(4, q),

=

“ P„¦+ (6, q) ∼ PSL(4, q),

=

“ P„¦’ (6, q) ∼ PSU(4, q);

=

(b) unexpected outer automorphisms of classical groups:

“ an automorphism of order 2 of PSp(4, q) for q even,

“ an automorphism of order 3 of P„¦+ (8, q);

(c) further simple groups:

“ Suzuki groups;

“ the groups G2 (q) and 3 D4 (q);

“ Ree groups.

In this section, we look at the geometric algebra underlying some of these phe-

nomena.

Notation: we use O+ (2mF) for the isometry group of the quadratic form of

Witt index m on a vector space of rank 2m (extending the notation over ¬nite

¬elds introduced earlier). We call this quadratic form Q hyperbolic. Moreover, the

¬‚at subspaces of rank 1 for Q are certain points in the corresponding projective

space PG(2m ’ 1, F); the set of such points is called a hyperbolic quadric in

PG(2m ’ 1, F).

We also denote the orthogonal group of the quadratic form

2

Q(x1 , . . . , x2m+1 ) = x1 x2 + · · · + x2m’1 x2m + x2m+1

by O(2m + 1, F), again in agreement with the ¬nite case.

76

7.1 Klein correspondence

The Klein correspondence relates the geometry of the vector space V = F 4 of

rank 4 over a ¬eld F with that of a vector space of rank 6 over F carrying a

quadratic form with Witt index 3.

It works as follows. Let W be the space of all 4 — 4 skew-symmetric matrices

over F. Then W has rank 6: the above-diagonal elements of such a matrix may be

chosen freely, and then the matrix is determined.

On the vector space W , there is a quadratic form Q given by

Q(X) = Pf(X) for all X ∈ W .

Recall the Pfaf¬an from Section 4.1, where we observed in particular that, if X =

(xi j ), then

Pf(X) = x12 x34 ’ x13 x24 + x14 x23 .

In particular, W is the sum of three hyperbolic planes, and the Witt index of Q is 3.

There is an action ρ of GL(4, F) on W given by the rule

ρ(P) : X ’ P XP

for P ∈ GL(4, F), X ∈ W . Now

Pf(PXP ) = det(P) Pf(X),

so ρ(P) preserves Q if and only if det(P) = 1. Thus ρ(SL(4, F)) ¤ O(Q), and

since SL(4, F) is equal to its derived group we have ρ(SL(4, F)) ¤ „¦+ (6, F).

It is easily checked that the kernel of ρ consists of scalars; so in fact we have

PSL(4, F) ¤ P„¦+ (6, F).

A calculation shows that in fact equality holds here. (More on this later.)

Theorem 7.1 P„¦+ (6, F) ∼ PSL(4, F).

=

Examining the geometry more closely throws more light on the situation.

Since the Pfaf¬an is the square root of the determinant, we have

Q(X) = 0 if and only if X is singular.

Now a skew-symmetric matrix has even rank; so if Q(X) = 0 but X = 0, then X

has rank 2.

77

Exercise 7.1 Any skew-symmetric n — n matrix of rank 2 has the form

X(v, w) = v w ’ w v

for some v, w ∈ F n .

Hint: Let B be such a matrix and let v and w span the row space of B. Then

B = x v + y w for some vectors x and y. Now by transposition we see that

x, y = v, w . Express x and y in terms of v and w, and use the skew-symmetry

to determine the coef¬cients up to a scalar factor.

Now X(v, w) = 0 if and only if v and w are linearly independent. If this holds,

then the row space is spanned by v and w. Moreover,

X(av + cw, bc + dw) = (ad ’ bc)X(v, w).

So there is a bijection between the rank 2 subspaces of F 4 and the ¬‚at subspaces

of W of rank 1. In terms of projective geometry, we have:

Proposition 7.2 There is a bijection between the lines of PG(3, F) and the points

on the hyperbolic quadric in PG(5, F), which intertwines the natural actions of

PSL(4, F) and P„¦+ (6, F).

This correspondence is called the Klein correspondence, and the quadric is

often referred to as the Klein quadric.

Now let A be a non-singular skew-symmetric matrix. The stabiliser of A in

ρ(SL(4, F)) consists of all matrices P such that PAP = A. These matrices com-

prise the symplectic group (see the exercise below). On the other hand, A is a

vector of W with Q(A) = 0, and so the stabiliser of A in the orthogonal group is

the 5-dimensional orthogonal group on A⊥ (where orthogonality is with respect to

the bilinear form obtained by polarising Q). Thus, we have

Theorem 7.3 P„¦(5, F) ∼ PSp(4, F).

=

Exercise 7.2 Let A be a non-singular skew-symmetric 4 — 4 matrix over a ¬eld F.

Prove that the following assertions are equivalent, for any vectors v, w ∈ F 4 :

(a) X(v, w) = v w ’ w v is orthogonal to A, with respect to the bilinear form

obtained by polarising the quadratic form Q(X) = Pf(X);

(b) v and w are orthogonal with respect to the symplectic form with matrix A† ,

that is, vA† w = 0.

78

Here the matrices A and A† are given by

« «

’a24

0 a12 a13 a14 0 a34 a23

¬ ’a12 a23 a24 · ¬ ’a34 ’a13 ·

0 0 a14

¬ ·, A =¬ ·.

†

A=

a34 a24 a12

’a13 ’a23 ’a14

0 0

’a14 ’a24 ’a34 0 ’a23 ’a12

a13 0

Now show that the transformation induced on W by a 4 — 4 matrix P ¬xes A if

and only if PA† P = A† , in other words, P is symplectic with respect to A† .

Note that, if A is the matrix of the standard symplectic form, then so is A† .

Now, we have two isomorphisms connecting the groups PSp(4, F) and P„¦(5, F)

in the case where F is a perfect ¬eld of characteristic 2. We can apply one and

then the inverse of the other to obtain an automorphism of the group PSp(4, F).

Now we show geometrically that it must be an outer automorphism.

The isomorphism in the preceding section was based on taking a vector space

of rank 5 and factoring out the radical Z. Recall that, on any coset Z + u, the

quadratic form takes each value in F precisely once; in particular, there is a unique

vector in each coset on which the quadratic form vanishes. Hence there is a bi-

jection between vectors in F 4 and vectors in F 5 on which the quadratic form

vanishes. This bijection is preserved by the isomorphism. Hence, under this iso-

morphism, the stabiliser of a point of the symplectic polar space is mapped to the

stabiliser of a point of the orthogonal polar space.

Now consider the isomorphism given by the Klein correspondence. Points on

the Klein quadric correspond to lines of PG(3, F), and it can be shown that, given

a non-singular matrix A, points of the Klein quadric orthogonal to A correspond

to ¬‚at lines with respect to the corresponding symplectic form on F 4 . In other

words, the isomorphism takes the stabiliser of a line (in the symplectic space) to

the stabiliser of a point (in the orthogonal space).

So the composition of one isomorphism with the inverse of the other inter-

changes the stabilisers of points and lines of the symplectic space, and so is an

outer automorphism of PSp(4, F).

7.2 The Suzuki groups

In certain cases, we can choose the outer automorphism such that its square is the

identity. Here is a brief account.

79

Theorem 7.4 Let F be a perfect ¬eld of characteristic 2. Then the polar space

de¬ned by a symplectic form on F 4 itself has a polarity if and only if F has an

automorphism σ satisfying σ2 = 2, where 2 denotes the automorphism x ’ x2 of

F.

Proof We take the standard symplectic form

B((x1 , x2 , x3 , x4 ), (y1 , y2 , y3 , y4 )) = x1 y2 + x2 y1 + x3 y4 + x4 y3 .

The Klein correspondence takes the line spanned by the two points (x1 , x2 , x3 , x4 )

and (y1 , y2 , y3 , y4 ) to the point wit coordinates zi j , for 1 ¤ i < j ¤ 4, where zi j =

xi y j + x j yi . This point lies on the Klein quadric with equation

z12 z34 + z13 z24 + z14 z23 = 0,

and also (if the line is ¬‚at) on the hyperplane z12 + z34 = 0. This hyperplane is

orthogonal to the point p with z12 = z34 = 1, zi j = 0 otherwise. Using coordinates

(z13 , z24 , z14 , z23 ) in p⊥ /p, we obtain a point of the symplectic space representing

the line. This gives the duality δ previously de¬ned.

Now take a point q = (a1 , a2 , a3 , a4 ) of the original space, and calculate its

image under the duality, by choosing two ¬‚at lines through q, calculating their

images, and taking the line joining them. Assuming that a1 and a4 are non-zero,

we can use the lines joining q to the points (a1 , a2 , 0, 0) and (0, a4 , a1 , 0); their

images are (a1 a3 , a2 a4 , a1 a4 , a2 a3 ) and (a2 , a2 , 0, a1 a2 + a3 a4 ). Now compute the

14

image of the line joining these two points, which turns out to be (a2 , a2 , a2 , a2 ). In

1234

all other cases, the result is the same. So δ 2 = 2.

If there is a ¬eld automorphism σ such that σ2 = 2, then δσ’1 is a duality

whose square is the identity, that is, a polarity.

Conversely, suppose that there is a polarity „. Then δ„ is a collineation, hence

a product of a linear transformation and a ¬eld automorphism, say δ„ = gσ. Since

δ2 = 2 and „2 = 1, we have that σ2 = 2 as required.

It can further be shown that the set of collineations which commute with this

polarity is a group G which acts doubly transitively on the set „¦ of absolute points

of the polarity, and that „¦ is an ovoid (that is, each ¬‚at line contains a unique point

of „¦. If |F| > 2, then the derived group of G is a simple group, the Suzuki group

Sz(F).

The ¬nite ¬eld GF(q), where q = 2m , has an automorphism σ satisfying σ2 = 2

if and only if m is odd (in which case, 2(m+1)/2 is the required automorphism). In

this case we have |„¦| = q2 + 1, and |Sz(q)| = (q2 + 1)q2 (q ’ 1). For q = 2, the

Suzuki group is not simple, being isomorphic to the Frobenius group of order 20.

80

7.3 Clifford algebras and spinors

We saw earlier (Proposition 3.11) that, if Q is a hyperbolic quadratic form on F 2m ,

then the maximal ¬‚at subspaces for Q fall into two families S + and S ’ , such that

if S and T are maximal ¬‚at subspaces, then S © T has even codimension in S and

T if and only if S and T belong to the same family.

In this section we represent the maximal ¬‚at subspaces as points in a larger

projective space, based on the space of spinors. The construction is algebraic.

First we brie¬‚y review facts about multilinear algebra.

Let V be a vector space over a ¬eld F, with rank m. The tensor algebra of V ,

N

written V , is the largest associative algebra generated by V in a linear fashion.

In other words,

O MO

n

V= V,

n≥0

N

where, for example, 2 V = V — V is spanned by symbols v — w, with v, w ∈ V ,

subject to the relations

(v1 + v2 ) — w = v1 — w + v2 — w,

v — (w1 + w2 ) = v — w1 + v — w2 ,

(cv) — w = c(v — w) = v — cw.

(Formally, it is the quotient of the free associative algebra over F with basis V

by the ideal generated by the differences of the left and right sides of the above

identities.) The algebra is N-graded, that is, it is a direct sum of components

N

Vn = n V indexed by the natural numbers, and Vn1 —Vn2 ⊆ Vn1 +n2 .

N

If (e1 , . . . , em ) is a basis for V , then a basis for n V consists of all symbols

ei1 — ei2 — · · · — ein ,

for i1 , . . . , in ∈ {1, . . . , m}; thus,

O

n

V ) = mn .

rk(

The exterior algebra of V is similarly de¬ned, but we add an additional con-

dition, namely v § v = 0 for all v ∈ V . (In this algebra we write the multiplication

V N

as §.) Thus, the exterior algebra V is the quotient of V by the ideal generated

by v — v for all v ∈ V .

81

In the exterior algebra, we have v § w = ’w § v. For

0 = (v + w) § (v + w) = v § v + v § w + w § v + w § w,

and the ¬rst and fourth terms on the right are zero. This means that, in any ex-

pression ei1 § ei2 § · · · § ein , we can rearrange the factors (possibly changing the

signs), and if two adjacent factors are equal then the product is zero. Thus, the nth

V

component n V has a basis consisting of all symbols

ei1 § ei2 § · · · § ein

where i1 < i2 < . . . < in . In particular,

^

n

m

V= ,

rk(

n

Vn

V + {0} for n > m; and

so that

^ m

m

‘ = 2m .

V) =

rk(

n

n=0

V

Note that rk( m V ) = 1. Any linear transformation T of V induces in a natural

N V

way a linear transformation on n V or n V for any n. In particular, the trans-

V V

formation m T induced on m V is a scalar, and this provides a coordinate-free

de¬nition of the determinant:

^

m

det(T ) = T.

Now let Q be a quadratic form on V . We de¬ne the Clifford algebra C(Q) of

Q to be the largest associative algebra generated by V in which the relation

v · v = Q(v)

holds. (We use · for the multiplication in C(Q). Note that, if Q is the zero form,

then C(Q) is just the exterior algebra. If B is the bilinear form obtained by polar-

ising Q, then we have

v · w + w · v = B(v, w).

This follows because

Q(v + w) = (v + w) · (v + w) = v · v + v · w + w · v + w · w

82

and also

Q(v + w) = Q(v) + Q(w) + B(v, w).

Now, when we arrange the factors in an expression like

ei1 · ei2 · · · ein ,

we obtain terms of degree n ’ 2 (and hence n ’ 4, n ’ 6, . . . as we continue). So

again we can say that the nth component has a basis consisting of all expressions

ei1 · ei2 · · · ein ,

where i1 < i2 < . . . < in , so that rk(C(Q)) = 2m . But this time the algebra is

not graded but only Z2 -graded. That is, if we let C0 and C1 be the sums of the

components of even (resp. odd) degree, then Ci ·C j ⊆ Ci+ j , where the superscripts

are taken modulo 2.

Suppose that Q polarises to a non-degenerate bilinear form B. Let G = O(Q)

and C = C(Q). The Clifford group “(Q) is de¬ned to be the group of all those

units s ∈ C such that s’1V s = V . Note that “(Q) has an action χ on V by the rule

s : v ’ s’1 vs.

Proposition 7.5 The action χ of “(Q) on V is orthogonal.

Proof

Q(s’1 vs) = (s’1 vs)2 = s’1 v2 s = s’1 Q(v)s = Q(v),

since Q(v), being a scalar, lies in the centre of C.

We state without proof:

(a) χ(“(Q)) = O(Q);

Proposition 7.6

(b) ker(χ) is the multiplicative group of invertible central elements of C(Q).

The structure of C(Q) can be calculated in special cases. The one which is of

interest to us is the following:

Theorem 7.7 Let Q be hyperbolic on F 2m . Then C(Q) ∼ End(S), where S is a

=

m over F called the space of spinors. In particular, “(Q)

vector space of rank 2

has a representation on S, called the spin representation.

83

The theorem follows immediately by induction from the following lemma:

Lemma 7.8 Suppose that Q = Q + yz, where y and z are variables not occurring

in Q . Then C(Q) ∼ M2—2 (C(Q )).

=

Proof Let V = V ⊥ e, f . Then V generates C(Q ), and V , e, f generate C(Q).

We represent these generators by 2 — 2 matrices over C(Q ) as follows:

v 0

v’ ,

’v

0

0 1

e’ ,

0 0

0 0

f’ .

1 0

Some checking is needed to establish the relations.

Let S be the vector space affording the spin representation. If U is a ¬‚at m-

subspace of V , let fU be the product of the elements in a basis of U. (Note that

fU is uniquely determined up to multiplication by non-zero scalars; indeed, the

subalgebra of C(Q) generated by U is isomorphic to the exterior algebra of U.)

Now it can be shown that C fU and fU C are minimal left and right ideals of C.

Since C ∼ End(S), each minimal left ideal has the form {T : V T ⊆ X} and each

=

minimal right ideal has the form {T : ker(T ) ⊇ Y }, where X and Y are subspaces

of V of dimension and codimension 1 respectively. In particular, a minimal left

ideal and a minimal right ideal intersect in a subspace of rank 1.

Thus we have a map σ from the set of ¬‚at m-subspaces of V into the set of

1-subspaces of S.

Vectors which span subspaces in the image of σ are called pure spinors.

Theorem 7.9 S = S+ • S’ , where rk(S+ ) = rk(S’ ) = 2m’1 . Moreover, any pure

spinor lies in either S+ or S’ according as the corresponding maximal ¬‚at sub-

space lies in S + or S ’ .

Furthermore, it is possible to de¬ne a quadratic form γ on S, whose corre-

sponding bilinear form β is non-degenerate, so that the following holds:

• if m is odd, then S+ and S’ are ¬‚at subspaces for γ, and β induces a non-

degenerate pairing between them;

84

• if m is even, then S+ and S’ are orthogonal with respect to β, and γ is

non-degenerate hyperbolic on each of them.

We now look brie¬‚y at the case m = 3. In this case, rk S+ = rk(S’ ) = 4. The

Clifford group has a subgroup of index 2 ¬xing S+ and S’ , and inducing dual

representations of SL(4, F) on them. We have here the Klein correspondence in

another form.

This case m = 4 is even more interesting, as we see in the next section.

7.4 Triality

Suppose that, in the notation of the preceding section, m = 4. That is, Q is a

hyperbolic quadratic form on V = F 8 , and the spinor space S is the direct sum of

two subspaces S+ and S’ of rank 8, each carrying a hyperbolic quadratic form of

rank 8. So each of these two spaces is isomorphic to the original space V . There

is an isomorphism „ (the triality map) of order 3 which takes V to S+ to S’ to V ,

and takes Q to γ|S+ to γ|S’ to Q. Moreover, „ induces an outer automorphism of

order 3 of the group P„¦+ (8, F).

Moreover, we have:

Proposition 7.10 A vector s ∈ S is a pure spinor if and only if

(a) s ∈ S+ or s ∈ S’ ; and

(b) γ(s) = 0.

Hence „ takes the stabiliser of a point to the stabiliser of a maximal ¬‚at sub-

space in S + to the stabiliser of a maximal ¬‚at subspace in S ’ back to the stabiliser

of a point.

It can be shown that the centraliser of „ in the orthogonal group is the group

G2 (F), an exceptional group of Lie type, which is the automorphism group of an

octonion algebra over F.

Further references for this chapter are in C. Chevalley, The Algebraic Theory

of Spinors and Clifford Algebras (Collected Works Vol. 2), Springer, 1997.

85

8 Further topics

The main topic in this section is Aschbacher™s Theorem, which describes the sub-

groups of the classical groups. First, there are two preliminaries: the O™Nan“Scott

Theorem, which does a similar job for the symmetric and alternating groups; and

the structure of extraspecial p-groups, which is an application of some of the ear-

lier material and also comes up unexpectedly in Aschbacher™s Theorem.

8.1 Extraspecial p-groups

An extraspecial p-group is a p-group (for some prime p) having the property that

its centre, derived group, and Frattini subgroup all coincide and have order p.

Otherwise said, it is a non-abelian p-group P with a normal subgroup Z such that

|Z| = p and P/Z is elementary abelian.

For example, of the ¬ve groups of order 8, two (the dihedral and quaternion

groups) are extraspecial; the other three are abelian.

Exercise 8.1 Prove that the above conditions are equivalent.

Theorem 8.1 An extraspecial p-group has order pm , where m is odd and greater

than 1. For any prime p and any odd m > 1, there are up to isomorphism exactly

two extraspecial p-groups of order pm .

Proof We translate the classi¬cation of extraspecial p-groups into geometric al-

gebra. First, note that such a group is nilpotent of class 2, and hence satis¬es the

following identities:

[xy, z] = [x, z][y, z], (2)

(xy)n = xn yn [y, x]n(n’1)/2 . (3)

(Here [x, y] = x’1 y’1 xy.)

Exercise 8.2 Prove that these equations hold in any group which is nilpotent of

class 2.

Let P be extraspecial with centre Z. Then Z is isomorphic to the additive group

of F = GF(p); we identify Z with F. Also, P/Z, being elementary abelian, is iso-

morphic to the additive group of a vector space V over F; we identify P/Z with V .

86

Of course, we have to be prepared to switch between additive and multiplicative

notation.

The structure of P is determined by two functions B : V —V ’ F and Q : V ’

F, de¬ned as follows. Since P/Z is elementary abelian, the commutator of any

two elements of P, or the pth power of any element of P, lie in Z. So commutation

and pth power are maps from P — P to F and from P to F. Each is unaffected by

changing its argument by an element of Z, since

[xz, y] = [x, y] = [x, yz] and (xz) p = x p

for z ∈ Z. So we have induced maps P/Z — P/Z ’ Z and P/Z ’ Z, which (under

the previous identi¬cations) are our required B and Q.

Exercise 8.3 Show that the structure of P can be reconstructed uniquely from the

¬eld F, the vector space V , and the maps B and Q above.

Now Equation (2) shows that B is bilinear. Since [x, x] = 1 for all x, it is

alternating. Elements of its radical lie in the centre of P, which is Z by assumption;

so B is nondegenerate. Thus B is a symplectic form.

In particular, n = rk(V ) is even; so |P| = pm where m = 1 + n is odd, proving

the ¬rst part of the theorem.

Now the analysis splits into two cases, according as p = 2 or p is odd.

Case p = 2 Now consider the map Q. Since |Z| = 2, we have [y, x] =

[x, y]’1 = [x, y] for all x, y. Now Equation (3) for n = 2, in additive notation,

asserts that

Q(x + y) = Q(x) + Q(y) + B(x, y),

In other words, Q is a quadratic form which polarises to B.

Since there are just two inequivalent quadratic forms, there are just two possi-

ble groups of each order up to isomorphism.

Case p odd The difference is caused by the behaviour of p(p ’ 1)/2 mod p:

for p odd, p divides p(p ’ 1)/2. Hence Equation (3) asserts

Q(x + y) = Q(x) + Q(y).

In other words, Q is linear. Any linear function can be uniquely represented as

Q(x) = B(x, a) for some vector a ∈ V . Since the symplectic group has just two

87

orbits on V , namely {0} and the set of all non-zero vectors, there are again just

two different groups. Note that the choice a = 0 gives a group of exponent p,

while a = 0 gives a group of exponent p2 .

Corollary 8.2 (a) The outer automorphism groups of the extraspecial 2-groups

of order 21+2r are the orthogonal groups „¦µ (2r, 2), for 3epsilon = ±1.

(b) Let p be odd. The outer automorphism group of the extraspecial p-group of

order p1+2r and exponent p is the general symplectic group GSp(2r, p) con-

sisting of linear maps preserving the symplectic form up to a scalar factor.

The automorphism group of the extraspecial p-group of order p1+2r and

exponent p2 is the stabiliser of a non-zero vector in the general symplectic

group.

Exercise 8.4 (a) Let P1 and P2 be groups and θ an isomorphism between cen-

tral subgroups Z1 and Z2 of P1 and P2 . The central product P1 —¦ P2 of P1 and

P2 with respect to θ is the factor group

(P1 — P2 )/{(z’1 , zθ) : z ∈ Z1 }.

Prove that the central product of extraspecial p-groups is extraspecial, and

corresponds to taking the orthogonal direct sum of the corresponding vector

spaces with forms.

(b) Hence prove that any extraspecial p-group of order p1+2r is a central prod-

uct of r extraspecial groups of order p3 where

“ if p = 2, all or all but one of the factors is dihedral;

“ if p is odd, all or all but one of the factors has exponent p.

We conclude with one more piece of information about extraspecial groups.

Let p be extraspecial of order p1+2r . The p elements of the centre lie in conjugacy

classes of size 1; all other conjugacy classes have size p, so there are p2r + p ’ 1

conjugacy classes. Hence there are the same number of irreducible characters.

But P/P has order p2r , so there are p2r characters of degree 1. It is easy to see

that the remaining p ’ 1 characters each have degree pr ; they are distinguished by

the values they take on the centre of P.

For p = 2, there is only one non-linear character, which is ¬xed by outer auto-

morphisms of P. Thus the representation of P lifts to the extension P.„¦µ (2r, 2).

88

For p = 2, suppose that P has exponent p. The subgroup Sp(2r, p) of the

outer automorphism group acts trivially on the centre, so ¬xes the p ’ 1 non-linear

representations; again, these representations lift to P. Sp(2r, p).

In the case of the last remark, the representation of P. Sp(2r, p) can be written

over GF(l) (l a prime power) provided that this ¬eld contains primitive pth roots

of unity, that is, l ≡ 1 (mod p). For the corresponding case with p = 2, we require

primitive 4th roots of unity, that is, l ≡ 1 (mod 4).

Thus, if these conditions hold, then GL(pr , l) contains a subgroup isomorphic

to P. Sp(2r, p) or P.„¦µ (2r, 2) (for p = 2).

8.2 The O™Nan“Scott Theorem

The O™Nan“Scott Theorem for subgroups of symmetric and alternating groups is

a slightly simpler prototype for Aschbacher™s Theorem

A group G is called almost simple if S ¤ G ¤ Aut(S) for some non-abelian

¬nite simple group S.

We de¬ne ¬ve classes of subgroups of the symmetric group Sn as follows:

C1 {Sk — Sl : k + l = n, k, l > 1} intransitive

C2 {Sk Sl : kl = n, k, l ≥ 2} imprimitive

C3 {Sk Sl : kl = n, k, l ≥ 2} product action

C4 {AGL(d, p) : p d = n} af¬ne

C5 {(T k ).(Out(T ) — S ) : k ≥ 2} diagonal

k

In the last row of the table, T is a non-abelian simple group, and the group

in question has its diagonal action: the stabiliser of a point is Aut(T ) — Sk =

(Td ).(Out(T ) — Sk ), where the embedding of Td in T k is the diagonal one, as

Td = {(t,t, . . . ,t) : t ∈ T },

and the action of T = Td is by inner automorphisms.

Now we can state the O™Nan“Scott Theorem.

Theorem 8.3 Let G be a subgroup of Sn or An , not equal to Sn or An . Then either

(a) G is contained in a subgroup belonging to one of the classes Ci , i = 1, . . . , 5;

or

(b) G is primitive and almost simple.

89

Note that the action of G in case (b) is not speci¬ed.

We sketch a proof of the theorem. If G is intransitive, then it is contained in

a maximal intransitive subgroup, which belongs to C1 . If G is transitive but im-

primitive, then it is contained in a maximal imprimitive subgroup, which belongs

to C2 . So we may suppose that G is primitive.

Let N be the socle of G, the product of its minimal normal subgroups. It is well

known and easy to prove that a primitive group has at most two minimal normal

subgroups; if there are two, then they are abelian. So N is a product of isomorphic

simple groups.

Now the steps required to complete the proof are as follows:

• If N is abelian, then it is elementary abelian of order pd for some prime p,

and N is regular, so n = pd . Then G ¤ AGL(d, p) = pd : GL(d, p), so G is

contained in a group in C4 .

• If N is non-abelian but not simple, then it can be shown that G is contained

in a group in C3 ∪ C5 .

• Of course, if N is simple, then G is almost simple.

In order to understand the maximal subgroups of Sn and An , there are two

things to do now. The theorem shows that the maximal subgroups are either in

the classes C1 “C5 or almost simple. First, we must resolve the question of which

of these groups contains another; this has been done by Liebeck, Praeger and

Saxl. Second, we must understand how almost simple groups act as primitive

permutation groups; equivalently, we must understand their maximal subgroups

(since a primitive action of a group is isomorphic to the action on the right cosets

of a maximal subgroup).

According to the Classi¬cation of Finite Simple Groups, most of the ¬nite

simple groups are classical groups. So this leads us naturally to the question of

proving a similar result for classical groups.

8.3 Aschbacher™s Theorem

Aschbacher™s Theorem is the required result. After a preliminary de¬nition, we

give the eight classes of subgroups, and then state the theorem.

A subgroup H of GL(n, F) is said to be irreducible if no subspace of F n is

invariant under H. We say that H is absolutely irreducible if, regarding elements

90

of H as n — n matrices over F, the group they generate is an irreducible subgroup

of GL(n, K) for any algebraic extension ¬eld K of F.

For example, the group

cos θ ’ sin θ

:θ∈R

SO(2, R) =

sin θ cos θ

is irreducible but not absolutely irreducible since, if we write it relative to the basis

(e1 + ie2 , e1 ’ ie2 ), the group would be

eiθ 0

.

e’iθ

0

Now we describe the Aschbacher classes. The examples of groups in these

classes will refer particularly to the general linear groups, but the de¬nitions apply

to all the classical groups. We let V be the natural module for the classical group

G.

C1 consists of reducible groups, those which stabilise a subspace W of V . In

GL(V ), the stabiliser of W consists of matrices which, in block form (the basis of

W coming ¬rst), have shape

AO

,

XB

where A ∈ GL(k, F), B ∈ GL(l, F) (with k +l = n), and X an arbitrary l —k matrix;

its structure is F kl : (GL(k, F) — GL(l, F)).

Note that, in a classical group with a sesquilinear form B, if the subspace W is

¬xed, then so is W ©W ⊥ . So we may assume that either W ©W ⊥ = {0} (so that

W is non-degenerate) or W ¤ W ⊥ (so that W is ¬‚at).

C2 consists of irreducible but imprimitive subgroups, those which preserve

a direct sum decomposition

V = V1 •V2 • · · · •Vt ,

where rk(Vi ) = m and n = mt; elements of the group permute these subspaces

among themselves. The stabiliser of the decomposition in GL(n, F) is GL(m, F)

St .

91

C3 consists of super¬eld groups. That is, a group in this class is a classi-

cal group acting on GF(qr )m , where rm = n, and it is embedded in GL(n, q) by

restricting scalars on the vector space from GF(qr ) to GF(q). Elements of the

Galois group of GF(qr ) over GF(q) are also linear. So in GL(n, q), a subgroup of

this form has shape GL(m, qr ) : Cr . For maximality, we may take r to be prime.

In the case of the classical group, we must sometimes modify the form (by

taking its trace from GF(qr ) to GF(q)); this may change the type of the form.

C4 consists of groups which preserve a tensor product structure V = F n1 —

F n2 , with n1 n2 = n. The appropriate subgroup of GL(n, F) is the central prod-

uct GL(n1 , F) —¦ GL(n2 , F). We can visualise this example most easily by taking

V to be the vector space of all n1 — n2 matrices, and letting the pair (A, B) ∈

GL(n1 , F) — GL(n2 , F) act by the rule

(A, B) : X ’ A’1 XB.

The kernel of the action is the appropriate subgroup which has to be factored out

to form the central product.

C5 consists of sub¬eld groups, that is, subgroups obtained by restricting the

matrix entries to a sub¬eld GF(q0 ) of GF(q), where q = qr (and we may take r to

0

be prime).

C6 consists of groups with extraspecial normal subgroups. We saw in the

section on extraspecial groups that the group P. Sp(2r, p) or (if p = 2) P.„¦µ (2r, 2)

can be embedded in GL(pr , l) if p (or 4) divides l ’ 1. These, together with the

scalars in GF(l), form the groups in this class.

C7 consists of groups preserving tensor decompositions of the form

V = V1 —V2 — · · · —Vt ,

with rk(Vi ) = m and n = mt . These are somewhat dif¬cult to visualise!

C8 consists of classical subgroups. Thus, any classical group acting on F n

can occur here as a subgroup of GL(n, F) provided that it is not obviously non-

maximal (e.g. we exclude „¦µ (2r, q) for q even, since these groups are contained

in Sp(2r, q). However, these groups would occur as class C8 subgroups of the

symplectic group.

92

Now some notation for Aschbacher™s Theorem. We let X(q) denote a clas-

sical group over GF(q), and V = GF(q)n its natural module. Also, „¦(q) is the

normal subgroup of X(q) such that „¦(q) modulo scalars is simple; and A(q)

is the normaliser of X(q) in the group of all invertible semilinear transforma-

tions of GF(q)n . A bar over the name of a group denotes that we have factored

out scalars. Note that A(q) is the automorphism group of „¦(q) except in the

cases X(q) = GL(n, q) (where there is an outer automorphism induced by dual-

ity), X(q) = O+ (8, q) (where there is an outer automorphism induced by triality),

and X(q) = Sp(4, q) with q even (where there is an outer automorphism induced

by the exceptional duality of the polar space).

Theorem 8.4 With the above notation, let „¦(q) ¤ G ¤ A(q), and suppose that H

is a subgroup of G not containing „¦(q). Then either

(a) H is contained in a subgroup in one of the classes C1 , . . . , C8 ; or

(b) H is absolutely irreducible and almost simple modulo scalars.

Kleidman and Liebeck, The Subgroup Structure of the Finite Classical Groups,

London Mathematical Society Lecture Note Series 129, Cambridge University

Press, 1990, gives further details, including an investigation of which of the groups

in the Aschbacher classes are actually maximal.

93

A short bibliography on classical groups

Standard books on classical groups are Artin [2], Dieudonn´ [14], Dickson [13]

e

and, for a more modern account, Taylor [22]. Cameron [5] describes the underly-

ing geometry.

Books on related topics include Cohn [10] on division rings, Gorenstein [15]

for the classi¬cation of ¬nite simple groups, the ATLAS [11] for properties of small

simple groups (including all the sporadic groups), the Handbook of Incidence

Geometry [4] for a detailed account of many topics including the geometry of

the classical groups, Chevalley [9] on Clifford algebras, spinors and triality, and

Kleidman and Liebeck [17] on subgroups of classical groups. (The last book is a

detailed commentary on the theorem of Aschbacher [3], itself the culmination of a

line of research commencing with Galois and continuing through Cooperstein [12]

and Kantor [16]. Cameron [6] has some geometric speculations on Aschbacher™s

Theorem.)

Carter [8] discusses groups of Lie type (identifying many of these with clas-

sical groups). The natural geometries for the groups of Lie type are buildings:

see Tits [23] for the classi¬cation of spherical buildings, and Scharlau [21] for a

modern account.

The other papers in the bibliography discuss aspects of the generation, sub-

groups, or representations of the classical groups. The list is not exhaustive!

References

[1] E. Artin, The orders of the classical simple groups, Comm. Pure Appl. Math.

8 (1955), 455“472.

[2] E. Artin, Geometric Algebra, Interscience, New York, 1957.

[3] M. Aschbacher, On the maximal subgroups of the ¬nite classical groups,

Invent. Math. 76 (1984), 469-514.

[4] F. Buekenhout (ed.), Handbook of Incidence Geometry, Elsevier, Amster-

dam, 1995.

[5] P. J. Cameron, Projective and Polar Spaces, QMW Maths Notes 13, London,

1991.

94

[6] P. J. Cameron, Finite geometry after Aschbacher™s Theorem: PGL(n, q) from

a Kleinian viewpoint, pp. 43“61 in Geometry, Combinatorics and Related

Topics (ed. J. W. P. Hirschfeld et al.), London Math. Soc. Lecture Notes 245,

Cambridge University Press, Cambridge, 1997.

[7] P. J. Cameron and W. M. Kantor, 2-transitive and anti¬‚ag transitive

collineation groups of ¬nite projective spaces, J. Algebra 60 (1979), 384“

422.

[8] R. W. Carter, Simple Groups of Lie Type, Wiley, New York, 1972.

[9] C. Chevalley, The Algebraic Theory of Spinors and Clifford Algebras (Col-

lected Works Vol. 2), Springer, Berlin, 1997.

[10] P. M. Cohn, Skew Field Constructions, London Math. Soc. Lecture Notes

27, Cambridge University Press, Cambridge, 1977.

[11] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, An

ATLAS of Finite Groups, Oxford University Press, Oxford, 1985.

[12] B. N. Cooperstein, Minimal degree for a permutation representation of a

classical group, Israel J. Math. 30 (1978), 213“235.

[13] L. E. Dickson, Linear Groups, with an Exposition of the Galois Field Theory,

Dover Publ. (reprint), New York, 1958.

[14] J. Dieudonn´ , La G´ ometrie des Groupes Classiques, Springer, Berlin, 1955.

e e

[15] D. Gorenstein, Finite Simple Groups: An Introduction to their Classi¬cation,

Plenum Press, New York, 1982.

[16] W. M. Kantor, Permutation representations of the ¬nite classical groups of

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