<< стр. 3(всего 3)СОДЕРЖАНИЕ

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 coefп¬Ѓcients 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= в€€ GL(2, F)
cd
corresponds the map

ux2 + vxy + wy2 в†’ 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
пЈ«2 пЈ¶
2
a 2ac c
ПЃ(A) = пЈ­ ab ad + bc cd пЈё в€€ GL(3, F).
b2 d2
2bd

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

(b) det(ПЃ(A)) = (det(A))3 .

(c) The quadratic form Q(u, v, w) = 4uw в€’ v2 is multiplied by a factor det(A)2
by the action of ПЃ(A).

Hence we п¬Ѓnd a subgroup of O(Q) which is isomorphic to SLВ± (2, F)/{В±I},
where SLВ± (2, F) is the group of matrices with determinant В±1. Moreover, its
intersection with SL(3, F) is SL(2, F)/{В±1}. In fact, these are the full groups
O(Q) and SO(Q) respectively.
We see that in this case,

Pв„¦(Q) в€ј 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 2Г—2 (F), the space of 2 Г— 2 matrices over F (where F is
any п¬Ѓeld). 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) Г— GL(2, F) on X, by the rule

ПЃ(A, B) : X в†’ Aв€’1 XB.

We see that ПЃ(A, B) preserves Q if and only if det(A) = det(B), and ПЃ(A, B) is the
identity if and only if A = B = О»I for some scalar О». So we have a subgroup of
O(Q) with the structure

((SL(2, F) Г— SL(2, F)) В· F Г— )/{(О»I, О»I) : О» в€€ F Г— }.

Moreover, the map T : X в†’ 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 в€’1 on V , while ПЃ(A, B)
has determinant equal to det(A)в€’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

Pв„¦(Q) в€ј PSL(2, F) Г— PSL(2, F)
=

if |F| > 3.

Exercise 6.5 Use the order formulae for п¬Ѓnite 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 п¬Ѓeld of characteristic 2. Let Q be a non-degenerate
quadratic form in n variables over F, where n is odd. Then O(Q) в€ј Sp(n в€’ 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 п¬Ѓxes each coset of Z. Since

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

and the map a в†’ a2 is a bijection of F, each coset of Z contains one vector with
each possible value of Q. Thus K = 1, and G в€ј 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 в€’ 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 Pв„¦(Q) with these
exceptions.
We conclude by constructing some more 2-transitive groups. Let F be a per-
fect п¬Ѓeld 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 вЂњsquare-seminilear
mapsвЂќ 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 п¬Ѓnite 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 + Оµ)/2
for a form of type Оµ.
Now specialise further to F = GF(2). In this case, вЂњsquare-semilinearвЂќ maps
are linear. So, given a quadratic form Q polarising to B, we have

Q (B) = {Q + L : L в€€ V в€— }.
Further, each linear form can be written as x в†’ B(x, a) for some п¬Ѓxed a в€€ V . Thus,
there is an O(Q)-invariant bijection between Q (B) and V . By WittвЂ™s Lemma,
O(Q) has just three orbits on V , namely

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

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

Q Оµ (B) \ {Q}, Q в€’Оµ (B),
{Q},

where Q has type Оµ and Q Оµ (B) is the set of all forms of type Оµ in Q (B).
It follows that Sp(2m, 2) acts 2-transitively on each of the two sets Q Оµ (B), with
cardinalities 2mв€’1 (2m + Оµ). The point stabiliser in these actions are OОµ (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 п¬Ѓrst 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 п¬Ѓeld 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 в†’ x в€’ Q(a)в€’1 B(x, a)a,

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

71
Proof Suppose that the transvection x в†’ 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 в€€ 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) =
в€’(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 в€€ 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 = в€’Q(a)в€’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 О»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 п¬Ѓnd 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 в†’ x + aB(x, v)u в€’ aB(x, u)v в€’ a2 Q(v)B(x, u)u
where Q(u) = B(u, v) = 0. The group of all such transformations for п¬Ѓxed 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 в€€ uвЉҐ
is abelian, and is isomorphic to the additive group of uвЉҐ / u .

72
Exercise 6.9 Write down the root subgroup Xu,v for the quadratic form Q(x1 , x2 , x3 ) =
2
x1 x3 в€’ x2 relative to to the given basis {e1 , e2 , e3 }, where u = e1 and v = e2 .

Now the details needed to apply IwasawaвЂ™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 Pв„¦(Q) в€ј PSp(4, 2) в€ј S6 .
= =

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

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

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

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

Note that the exception of the case of rank 4 and Witt index 2 is really necessary
for (d): the group в„¦(Q) п¬Ѓxes 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 IwasawaвЂ™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 Pв„¦(Q) is simple.

It remains for us to discover the order of Pв„¦(Q) over a п¬Ѓnite п¬Ѓeld. 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)/в„¦(Q) в€ј F Г— /(F Г— )2 .
=

(b) Let F be a perfect п¬Ѓeld 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) :
в„¦(Q)| = 2.

73
The proof of part (a) involves deп¬Ѓning a homomorphism from SO(Q) to F Г— /(F Г— )2
called the spinor norm, and showing that it is onto and its kernel is в„¦(Q) except
in the excluded case.
In the remaining cases, we work over the п¬Ѓnite п¬Ѓeld GF(q), and write O(n, q),
understanding that if n is even then OОµ (n, q) is meant.

Proposition 6.5 Excluding the case q even and n odd:
(a) | SO(n, q) : в„¦(n, q)| = 2.

(b) For q odd, в€’I в€€ в„¦Оµ (2m, q) if and only if qm в€ј Оµ (mod 4).
=

The last part is proved by calculating the spinor norm of в€’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 в‰Ґ 2, excluding the case Pв„¦+ (4, 2), we have
mв€’1
(q в€’ Оµ) в€Џ (q2i в€’ 1)
Оµ
(4, qm в€’ Оµ),
m(mв€’1) m
| Pв„¦ (2m, q)| = q
i=1
m
в€Џ(q2i в€’ 1)
m2
| Pв„¦(2m + 1, q)| = (2, q в€’ 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 в„¦(n, q), and another 2 on
factoring out the scalars if and only if 4 divides qm в€’ Оµ. For q even, | SO(n, q)| is
twice the bracketed expression, and we lose the factor 2 on passing to в„¦(n, q) =
Pв„¦(n, q).

Now we note that | Pв„¦(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 в‰Ґ 3 and q odd.
This follows from the result of the following exercise.

Exercise 6.10 Let q be odd and m в‰Ґ 2.
(a) The group PSp(2m, q) has m/2 + 1 conjugacy classes of elements of or-
der 2.

74
(b) The group Pв„¦(2m + 1, q) has m conjugacy classes of elements of order 2.

Hint: if t в€€ Sp(2m, q) or t в€€ в„¦(2m + 1, q) = Pв„¦(2m + 1, q) satisп¬Ѓes t 2 = 1, then
V = V + вЉ• V в€’ , where vt = О»v for v в€€ V О» ; and the subspaces V + and V в€’ are or-
thogonal. Show that there are m possibilities for the subspaces V + and V в€’ up to
isometry; in the symplectic case, replacing t by в€’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 в€€ Sp(2m, q) with t 2 = в€’1.

It follows from the Classiп¬Ѓcation 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) в€ј Pв„¦(2m + 1, q) for m в‰Ґ 3, q odd
=

and
PSL(3, 4) в€ј PSL(4, 2) в€ј A8 .
= =
The lecture course will not contain detailed proofs of the simplicity of Pв„¦(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 + В· В· В· + x2mв€’1 x2m

of Witt index m in 2m variables, the п¬‚at m-spaces fall into two families F + and
F в€’ , with the property that the intersection of two п¬‚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 п¬Ѓx 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 п¬Ѓxes all the others. So SO+ (2m, q) has a subgroup of index 2 п¬Ѓx-
ing the two families, which is в„¦+ (2m, q). (In the case where q is odd, such a
transformation has determinant в€’1.)

75
7 Klein correspondence and triality
The orthogonal groups in dimension up to 8 have some remarkable properties.
These include, in the п¬Ѓnite case,

(a) isomorphisms between classical groups:

вЂ“ Pв„¦в€’ (4, q) в€ј PSL(2, q2 ),
=
вЂ“ Pв„¦(5, q) в€ј PSp(4, q),
=
вЂ“ Pв„¦+ (6, q) в€ј PSL(4, q),
=
вЂ“ Pв„¦в€’ (6, q) в€ј PSU(4, q);
=

(b) unexpected outer automorphisms of classical groups:

вЂ“ an automorphism of order 2 of PSp(4, q) for q even,
вЂ“ an automorphism of order 3 of Pв„¦+ (8, q);

(c) further simple groups:

вЂ“ Suzuki groups;
вЂ“ the groups G2 (q) and 3 D4 (q);
вЂ“ 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 п¬Ѓnite
п¬Ѓelds introduced earlier). We call this quadratic form Q hyperbolic. Moreover, the
п¬‚at subspaces of rank 1 for Q are certain points in the corresponding projective
space PG(2m в€’ 1, F); the set of such points is called a hyperbolic quadric in
PG(2m в€’ 1, F).
We also denote the orthogonal group of the quadratic form
2
Q(x1 , . . . , x2m+1 ) = x1 x2 + В· В· В· + x2mв€’1 x2m + x2m+1

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

76
7.1 Klein correspondence
The Klein correspondence relates the geometry of the vector space V = F 4 of
rank 4 over a п¬Ѓeld 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 Г— 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 в€€ W .

Recall the Pfafп¬Ѓan from Section 4.1, where we observed in particular that, if X =
(xi j ), then
Pf(X) = x12 x34 в€’ 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 ПЃ of GL(4, F) on W given by the rule

ПЃ(P) : X в†’ P XP

for P в€€ GL(4, F), X в€€ W . Now

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

so ПЃ(P) preserves Q if and only if det(P) = 1. Thus ПЃ(SL(4, F)) в‰¤ O(Q), and
since SL(4, F) is equal to its derived group we have ПЃ(SL(4, F)) в‰¤ в„¦+ (6, F).
It is easily checked that the kernel of ПЃ consists of scalars; so in fact we have
PSL(4, F) в‰¤ Pв„¦+ (6, F).
A calculation shows that in fact equality holds here. (More on this later.)

Theorem 7.1 Pв„¦+ (6, F) в€ј PSL(4, F).
=

Examining the geometry more closely throws more light on the situation.
Since the Pfafп¬Ѓan 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 Г— n matrix of rank 2 has the form

X(v, w) = v w в€’ w v

for some v, w в€€ 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 coefп¬Ѓcients 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 в€’ bc)X(v, w).

So there is a bijection between the rank 2 subspaces of F 4 and the п¬‚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 Pв„¦+ (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
ПЃ(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 AвЉҐ (where orthogonality is with respect to
the bilinear form obtained by polarising Q). Thus, we have

Theorem 7.3 Pв„¦(5, F) в€ј PSp(4, F).
=

Exercise 7.2 Let A be a non-singular skew-symmetric 4 Г— 4 matrix over a п¬Ѓeld F.
Prove that the following assertions are equivalent, for any vectors v, w в€€ F 4 :
(a) X(v, w) = v w в€’ 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 AвЂ  ,
that is, vAвЂ  w = 0.

78
Here the matrices A and AвЂ  are given by
пЈ« пЈ¶ пЈ« пЈ¶
в€’a24
0 a12 a13 a14 0 a34 a23
пЈ¬ в€’a12 a23 a24 пЈ· пЈ¬ в€’a34 в€’a13 пЈ·
0 0 a14
пЈ¬ пЈ·, A =пЈ¬ пЈ·.
вЂ
A=пЈ­
a34 пЈё пЈ­ a24 a12 пЈё
в€’a13 в€’a23 в€’a14
0 0
в€’a14 в€’a24 в€’a34 0 в€’a23 в€’a12
a13 0

Now show that the transformation induced on W by a 4 Г— 4 matrix P п¬Ѓxes A if
and only if PAвЂ  P = AвЂ  , in other words, P is symplectic with respect to AвЂ  .
Note that, if A is the matrix of the standard symplectic form, then so is AвЂ  .

Now, we have two isomorphisms connecting the groups PSp(4, F) and Pв„¦(5, F)
in the case where F is a perfect п¬Ѓeld 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 п¬‚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 п¬Ѓeld of characteristic 2. Then the polar space
deп¬Ѓned by a symplectic form on F 4 itself has a polarity if and only if F has an
automorphism Пѓ satisfying Пѓ2 = 2, where 2 denotes the automorphism x в†’ 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 в‰¤ i < j в‰¤ 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 п¬‚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 pвЉҐ /p, we obtain a point of the symplectic space representing
the line. This gives the duality Оґ previously deп¬Ѓned.
Now take a point q = (a1 , a2 , a3 , a4 ) of the original space, and calculate its
image under the duality, by choosing two п¬‚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 Оґ 2 = 2.

If there is a п¬Ѓeld automorphism Пѓ such that Пѓ2 = 2, then ОґПѓв€’1 is a duality
whose square is the identity, that is, a polarity.
Conversely, suppose that there is a polarity П„. Then ОґП„ is a collineation, hence
a product of a linear transformation and a п¬Ѓeld automorphism, say ОґП„ = gПѓ. Since
Оґ2 = 2 and П„2 = 1, we have that Пѓ2 = 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 в„¦ of absolute points
of the polarity, and that в„¦ is an ovoid (that is, each п¬‚at line contains a unique point
of в„¦. If |F| > 2, then the derived group of G is a simple group, the Suzuki group
Sz(F).
The п¬Ѓnite п¬Ѓeld GF(q), where q = 2m , has an automorphism Пѓ satisfying Пѓ2 = 2
if and only if m is odd (in which case, 2(m+1)/2 is the required automorphism). In
this case we have |в„¦| = q2 + 1, and |Sz(q)| = (q2 + 1)q2 (q в€’ 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 п¬‚at subspaces for Q fall into two families S + and S в€’ , such that
if S and T are maximal п¬‚at subspaces, then S в€© 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 п¬‚at subspaces as points in a larger
projective space, based on the space of spinors. The construction is algebraic.
First we brieп¬‚y review facts about multilinear algebra.
Let V be a vector space over a п¬Ѓeld 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,
nв‰Ґ0
N
where, for example, 2 V = V вЉ— V is spanned by symbols v вЉ— w, with v, w в€€ V ,
subject to the relations

(v1 + v2 ) вЉ— w = v1 вЉ— w + v2 вЉ— w,
v вЉ— (w1 + w2 ) = v вЉ— w1 + v вЉ— w2 ,
(cv) вЉ— w = c(v вЉ— w) = v вЉ— 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 вЉ—Vn2 вЉ† Vn1 +n2 .
N
If (e1 , . . . , em ) is a basis for V , then a basis for n V consists of all symbols

ei1 вЉ— ei2 вЉ— В· В· В· вЉ— ein ,

for i1 , . . . , in в€€ {1, . . . , m}; thus,
O
n
V ) = mn .
rk(

The exterior algebra of V is similarly deп¬Ѓned, but we add an additional con-
dition, namely v в€§ v = 0 for all v в€€ V . (In this algebra we write the multiplication
V N
as в€§.) Thus, the exterior algebra V is the quotient of V by the ideal generated
by v вЉ— v for all v в€€ V .

81
In the exterior algebra, we have v в€§ w = в€’w в€§ v. For

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

and the п¬Ѓrst and fourth terms on the right are zero. This means that, in any ex-
pression ei1 в€§ ei2 в€§ В· В· В· в€§ 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 в€§ ei2 в€§ В· В· В· в€§ ein

where i1 < i2 < . . . < in . In particular,
^
n
m
V= ,
rk(
n
Vn
V + {0} for n > m; and
so that
^ m
m
в€‘ = 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
deп¬Ѓnition of the determinant:
^
m
det(T ) = T.

Now let Q be a quadratic form on V . We deп¬Ѓne the Clifford algebra C(Q) of
Q to be the largest associative algebra generated by V in which the relation

v В· v = Q(v)

holds. (We use В· 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 В· w + w В· v = B(v, w).
This follows because

Q(v + w) = (v + w) В· (v + w) = v В· v + v В· w + w В· v + w В· 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 В· ei2 В· В· В· ein ,

we obtain terms of degree n в€’ 2 (and hence n в€’ 4, n в€’ 6, . . . as we continue). So
again we can say that the nth component has a basis consisting of all expressions

ei1 В· ei2 В· В· В· 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 В·C j вЉ† 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 О“(Q) is deп¬Ѓned to be the group of all those
units s в€€ C such that sв€’1V s = V . Note that О“(Q) has an action П‡ on V by the rule

s : v в†’ sв€’1 vs.

Proposition 7.5 The action П‡ of О“(Q) on V is orthogonal.
Proof

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

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

We state without proof:
(a) П‡(О“(Q)) = O(Q);
Proposition 7.6

(b) ker(П‡) 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) в€ј End(S), where S is a
=
m over F called the space of spinors. In particular, О“(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) в€ј M2Г—2 (C(Q )).
=

Proof Let V = V вЉҐ e, f . Then V generates C(Q ), and V , e, f generate C(Q).
We represent these generators by 2 Г— 2 matrices over C(Q ) as follows:

v 0
vв†’ ,
в€’v
0
0 1
eв†’ ,
0 0
0 0
fв†’ .
1 0

Some checking is needed to establish the relations.

Let S be the vector space affording the spin representation. If U is a п¬‚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 в€ј End(S), each minimal left ideal has the form {T : V T вЉ† X} and each
=
minimal right ideal has the form {T : ker(T ) вЉ‡ 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 Пѓ from the set of п¬‚at m-subspaces of V into the set of
1-subspaces of S.
Vectors which span subspaces in the image of Пѓ are called pure spinors.

Theorem 7.9 S = S+ вЉ• Sв€’ , where rk(S+ ) = rk(Sв€’ ) = 2mв€’1 . Moreover, any pure
spinor lies in either S+ or Sв€’ according as the corresponding maximal п¬‚at sub-
space lies in S + or S в€’ .

Furthermore, it is possible to deп¬Ѓne a quadratic form Оі on S, whose corre-
sponding bilinear form ОІ is non-degenerate, so that the following holds:

вЂў if m is odd, then S+ and Sв€’ are п¬‚at subspaces for Оі, and ОІ induces a non-
degenerate pairing between them;

84
вЂў if m is even, then S+ and Sв€’ are orthogonal with respect to ОІ, and Оі is
non-degenerate hyperbolic on each of them.

We now look brieп¬‚y at the case m = 3. In this case, rk S+ = rk(Sв€’ ) = 4. The
Clifford group has a subgroup of index 2 п¬Ѓxing S+ and Sв€’ , 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 Sв€’ 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 П„ (the triality map) of order 3 which takes V to S+ to Sв€’ to V ,
and takes Q to Оі|S+ to Оі|Sв€’ to Q. Moreover, П„ induces an outer automorphism of
order 3 of the group Pв„¦+ (8, F).
Moreover, we have:

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

(a) s в€€ S+ or s в€€ Sв€’ ; and

(b) Оі(s) = 0.

Hence П„ takes the stabiliser of a point to the stabiliser of a maximal п¬‚at sub-
space in S + to the stabiliser of a maximal п¬‚at subspace in S в€’ back to the stabiliser
of a point.
It can be shown that the centraliser of П„ 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 AschbacherвЂ™s Theorem, which describes the sub-
groups of the classical groups. First, there are two preliminaries: the OвЂ™NanвЂ“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 AschbacherвЂ™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 п¬Ѓve 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 classiп¬Ѓcation of extraspecial p-groups into geometric al-
gebra. First, note that such a group is nilpotent of class 2, and hence satisп¬Ѓes the
following identities:

[xy, z] = [x, z][y, z], (2)
(xy)n = xn yn [y, x]n(nв€’1)/2 . (3)

(Here [x, y] = xв€’1 yв€’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 Г—V в†’ F and Q : V в†’
F, deп¬Ѓned 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 Г— 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 в€€ Z. So we have induced maps P/Z Г— P/Z в†’ Z and P/Z в†’ Z, which (under
the previous identiп¬Ѓcations) are our required B and Q.

Exercise 8.3 Show that the structure of P can be reconstructed uniquely from the
п¬Ѓeld 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 п¬Ѓrst 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]в€’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 в€’ 1)/2 mod p:
for p odd, p divides p(p в€’ 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 в€€ 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 в„¦Оµ (2r, 2), for 3epsilon = В±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 Оё an isomorphism between cen-
tral subgroups Z1 and Z2 of P1 and P2 . The central product P1 в—¦ P2 of P1 and
P2 with respect to Оё is the factor group

(P1 Г— P2 )/{(zв€’1 , zОё) : z в€€ 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

вЂ“ if p = 2, all or all but one of the factors is dihedral;
вЂ“ 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 в€’ 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 в€’ 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 п¬Ѓxed by outer auto-
morphisms of P. Thus the representation of P lifts to the extension P.в„¦Оµ (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 п¬Ѓxes the p в€’ 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 п¬Ѓeld contains primitive pth roots
of unity, that is, l в‰Ў 1 (mod p). For the corresponding case with p = 2, we require
primitive 4th roots of unity, that is, l в‰Ў 1 (mod 4).
Thus, if these conditions hold, then GL(pr , l) contains a subgroup isomorphic
to P. Sp(2r, p) or P.в„¦Оµ (2r, 2) (for p = 2).

8.2 The OвЂ™NanвЂ“Scott Theorem
The OвЂ™NanвЂ“Scott Theorem for subgroups of symmetric and alternating groups is
a slightly simpler prototype for AschbacherвЂ™s Theorem
A group G is called almost simple if S в‰¤ G в‰¤ Aut(S) for some non-abelian
п¬Ѓnite simple group S.
We deп¬Ѓne п¬Ѓve classes of subgroups of the symmetric group Sn as follows:
C1 {Sk Г— Sl : k + l = n, k, l > 1} intransitive
C2 {Sk Sl : kl = n, k, l в‰Ґ 2} imprimitive
C3 {Sk Sl : kl = n, k, l в‰Ґ 2} product action
C4 {AGL(d, p) : p d = n} afп¬Ѓne
C5 {(T k ).(Out(T ) Г— S ) : k в‰Ґ 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 ) Г— Sk =
(Td ).(Out(T ) Г— Sk ), where the embedding of Td in T k is the diagonal one, as

Td = {(t,t, . . . ,t) : t в€€ T },

and the action of T = Td is by inner automorphisms.
Now we can state the OвЂ™NanвЂ“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 speciп¬Ѓed.
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:

вЂў 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 в‰¤ AGL(d, p) = pd : GL(d, p), so G is
contained in a group in C4 .

вЂў If N is non-abelian but not simple, then it can be shown that G is contained
in a group in C3 в€Є C5 .

вЂў 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 вЂ“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 Classiп¬Ѓcation of Finite Simple Groups, most of the п¬Ѓnite
simple groups are classical groups. So this leads us naturally to the question of
proving a similar result for classical groups.

8.3 AschbacherвЂ™s Theorem
AschbacherвЂ™s Theorem is the required result. After a preliminary deп¬Ѓnition, 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 Г— n matrices over F, the group they generate is an irreducible subgroup
of GL(n, K) for any algebraic extension п¬Ѓeld K of F.
For example, the group

cos Оё в€’ sin Оё
:Оёв€€R
SO(2, R) =
sin Оё cos Оё

is irreducible but not absolutely irreducible since, if we write it relative to the basis
(e1 + ie2 , e1 в€’ ie2 ), the group would be

eiОё 0
.
eв€’iОё
0

Now we describe the Aschbacher classes. The examples of groups in these
classes will refer particularly to the general linear groups, but the deп¬Ѓnitions 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 п¬Ѓrst), have shape
AO
,
XB
where A в€€ GL(k, F), B в€€ GL(l, F) (with k +l = n), and X an arbitrary l Г—k matrix;
its structure is F kl : (GL(k, F) Г— GL(l, F)).
Note that, in a classical group with a sesquilinear form B, if the subspace W is
п¬Ѓxed, then so is W в€©W вЉҐ . So we may assume that either W в€©W вЉҐ = {0} (so that
W is non-degenerate) or W в‰¤ W вЉҐ (so that W is п¬‚at).

C2 consists of irreducible but imprimitive subgroups, those which preserve
a direct sum decomposition

V = V1 вЉ•V2 вЉ• В· В· В· вЉ•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 superп¬Ѓeld 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 вЉ—
F n2 , with n1 n2 = n. The appropriate subgroup of GL(n, F) is the central prod-
uct GL(n1 , F) в—¦ GL(n2 , F). We can visualise this example most easily by taking
V to be the vector space of all n1 Г— n2 matrices, and letting the pair (A, B) в€€
GL(n1 , F) Г— GL(n2 , F) act by the rule
(A, B) : X в†’ Aв€’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 subп¬Ѓeld groups, that is, subgroups obtained by restricting the
matrix entries to a subп¬Ѓeld 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.в„¦Оµ (2r, 2)
can be embedded in GL(pr , l) if p (or 4) divides l в€’ 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 вЉ—V2 вЉ— В· В· В· вЉ—Vt ,
with rk(Vi ) = m and n = mt . These are somewhat difп¬Ѓcult 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 в„¦Оµ (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 AschbacherвЂ™s Theorem. We let X(q) denote a clas-
sical group over GF(q), and V = GF(q)n its natural module. Also, в„¦(q) is the
normal subgroup of X(q) such that в„¦(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 в„¦(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 в„¦(q) в‰¤ G в‰¤ A(q), and suppose that H
is a subgroup of G not containing в„¦(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 , DieudonnВґ , Dickson 
e
and, for a more modern account, Taylor . Cameron  describes the underly-
ing geometry.
Books on related topics include Cohn  on division rings, Gorenstein 
for the classiп¬Ѓcation of п¬Ѓnite simple groups, the ATLAS  for properties of small
simple groups (including all the sporadic groups), the Handbook of Incidence
Geometry  for a detailed account of many topics including the geometry of
the classical groups, Chevalley  on Clifford algebras, spinors and triality, and
Kleidman and Liebeck  on subgroups of classical groups. (The last book is a
detailed commentary on the theorem of Aschbacher , itself the culmination of a
line of research commencing with Galois and continuing through Cooperstein 
and Kantor . Cameron  has some geometric speculations on AschbacherвЂ™s
Theorem.)
Carter  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  for the classiп¬Ѓcation of spherical buildings, and Scharlau  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
 E. Artin, The orders of the classical simple groups, Comm. Pure Appl. Math.
8 (1955), 455вЂ“472.

 E. Artin, Geometric Algebra, Interscience, New York, 1957.

 M. Aschbacher, On the maximal subgroups of the п¬Ѓnite classical groups,
Invent. Math. 76 (1984), 469-514.

 F. Buekenhout (ed.), Handbook of Incidence Geometry, Elsevier, Amster-
dam, 1995.

 P. J. Cameron, Projective and Polar Spaces, QMW Maths Notes 13, London,
1991.

94
 P. J. Cameron, Finite geometry after AschbacherвЂ™s Theorem: PGL(n, q) from
a Kleinian viewpoint, pp. 43вЂ“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.

 P. J. Cameron and W. M. Kantor, 2-transitive and antiп¬‚ag transitive
collineation groups of п¬Ѓnite projective spaces, J. Algebra 60 (1979), 384вЂ“
422.

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

 C. Chevalley, The Algebraic Theory of Spinors and Clifford Algebras (Col-
lected Works Vol. 2), Springer, Berlin, 1997.

 P. M. Cohn, Skew Field Constructions, London Math. Soc. Lecture Notes
27, Cambridge University Press, Cambridge, 1977.

 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.

 B. N. Cooperstein, Minimal degree for a permutation representation of a
classical group, Israel J. Math. 30 (1978), 213вЂ“235.

 L. E. Dickson, Linear Groups, with an Exposition of the Galois Field Theory,
Dover Publ. (reprint), New York, 1958.

 J. DieudonnВґ , La GВґ ometrie des Groupes Classiques, Springer, Berlin, 1955.
e e

 D. Gorenstein, Finite Simple Groups: An Introduction to their Classiп¬Ѓcation,
Plenum Press, New York, 1982.

 W. M. Kantor, Permutation representations of the п¬Ѓnite classical groups of
small degree or rank, J. Algebra 60 (1979), 158вЂ“168.

 P. B. Kleidman and M. W. Liebeck, The Subgroup Structure of the Finite
Classical Groups, London Math. Soc. Lecture Notes 129, Cambridge Univ.
Press, Cambridge, 1990.

 M. W. Liebeck, On the orders of maximal subgroups of the п¬Ѓnite classical
groups, Proc. London Math. Soc. (3) 50 (1985), 426вЂ“446.

95
 G. Malle, J. Saxl and T. Weigel, Generation of classical groups, Geom. Ded-
icata 49 (1994), 85вЂ“116.

ВЁ
 H. MВЁ urer, Eine Charakterisierung der Permutationsgruppe PSL(2, K) uber
a
einem quadratisch abgeschlossenen KВЁ rper K der Charakteristik = 2, Geom.
o
Dedicata 36 (1990), 235вЂ“237.

 R. Scharlau, Buildings, pp. 477вЂ“645 in Handbook of Incidence Geometry (F.
Buekenhout, ed.), Elsevier, Amsterdam, 1995.

 D. E. Taylor, The Geometry of the Classical Groups, Heldermann Verlag,
Berlin, 1992.

 J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in
Math. 382, SpringerвЂ“Verlag, Berlin, 1974.

96

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