O L I V E R K I N G

ON S U B G R O U P S OF T H E S P E C I A L L I N E A R G R O U P

C O N T A I N I N G T H E S P E C I A L U N I T A R Y G R O U P

I N T R O D U C T I O N

In [4], Dye proved the maximality in the special linear group of its intersec- tion with the general symplectic group, and in [9], I proved the analogous result for the general orthogonal group (in which case it is necessary to restrict attention to fields of characteristic not 2). In this paper we consider the intersection of the general unitary group with the special linear group. As in [4], [8] we get a maximal subgroup that may be thought of as the stabilizer of a geometric configuration, namely the set of 'isotropic' 1- dimensional subspaces (Proposition 1). We also determine the subgroups of the special linear group that contain the special unitary group.

Let V be a vector space of dimension n ~> 2, defined over a field K that has a non-trivial involutory automorphism; the automorphic image of 2 ~ K will be written )~. The elements of K left fixed by the automorphism form a subfield Ko of K, and K is a separable normal extension of Ko of degree 2. As usual, let GL,(K) and SL,(K) be the general and special linear groups on V. Let C be a non-degenerate hermitian form on V of Witt index v >~ 1, thus

and

C(x, ;~y + uz) = 2C(x, y) + uC(x, z),

c(,~x + uy, z) = £C(x, z) + ~c(y, z),

C(y, x) = C(x, y) Vx, y, z s V, V2, p e K ;

we shall usually write (x, y) for C(x, y). Let U,(K), SU,(K) and GU,(K) be the unitary, special unitary and general unitary groups of C; adapting notation from [4], we denote GU.(K) n SL,(K) by SGU,(K) and the quo- tient of SGU,(K) by its centre by PSGU,(K). We follow standard practice in writing, for example, SL,(K) = SL,(q) when K = GF(q). Our main result is

THEOREM I. Any proper subgroup of SL,(K) containin9 SU.(K) lies in

SGU,(K), except when n = 2 and K = GF(9). A proper subgroup o f SLz(9) containin 9 SU2(9) either lies in SGU2(9 ) or, when factored out by its centre, is

isomorphic to A 5 .

Geometriae Dedicata 19 (1985) 297-310. 0046-5755/85.15. © 1985 by D. Reidel Publishin9 Company.

298 O L I V E R K I N G

Theorem I is proved in Sections 2 and 3, the cases n >~ 3 and n = 2 being considered separately because of significant differences in the proofs. An immediate corollary is that SGU,(K) is maximal in SL,(K). In the case n = 2, it is convenient to use the well-known isomorphism between S Uz(K )

and SL2(Ko) (cf. I-3, p. 46]). Our approach is geometrical in nature and uses only the well-known fact that SL,(K) is generated by its transvections. In Section 4 we determine the subgroups of SL,(K) containing SU,(K); we then state conditions for the maximality of GU,(K), SU,(K) and PSU,(K) in GL,(K), SL,(K) and PSL,(K), respectively, and interpret these conditions when K -- C and when K is finite.

For finite fields, the cases n = 2, 3 of Theorem I are known (Results 1, 2), and the other cases may be deduced from Wagner [14] (except when K = GF(4)), from Kantor [-7] or from Piper [-11], 1-12], given the early part of our proof.

1. F U R T H E R N O T A T I O N AND P R E L I M I N A R Y R E S U L T S

Our notation mostly follows [8]. Given a subspace W of V, its conjugate will be denoted by W'. The subspace W is said to be totally isotropic if W _ W', non-isotropic if W c~ W' = {0} and anisotropic if it has no totally isotropic 1-dimensional subspaces; as W c~ W' is totally isotropic for any subspace W, every anisotropic subspace is non-isotropic. A vector x is said to be isotropic [-resp., non-isotropic] if the subspace (x ) is totally isotropic [resp., non-isotropic]; a totally isotropic 1-dimensional subspace will also be called isotropic. If W is non-isotropic, then we denote by SU(W) the subgroup of SU,(K) consisting of all the elements fixing each vector in W';

SU(W) is then isomorphic to the special unitary group of C restricted to W.

The following results are stated in terms of our notation.

RESULT 1 (Dickson [2]). I f K = GF(q2), then any proper subgroup of

SL2(q 2) containing SU2(q 2) lies in SGU2(q2), except when n = 2 and q = 3. A

proper subgroup of SL2(9) containing SU2(9) either lies in SGU2(9) or, when

factored out by its centre, is isomorphic to As •

Dickson actually lists the subgroups of PSL2(q) for all q, but as SU2(q 2) contains the centre of SLz(qZ), the subgroups of SLa(q 2) containing SU2(q 2) correspond to the subgroups of PSLz(q z) containing PSUa(q2). The iso- morphism between SUa(K) and SL2(Ko) referred to above extends to an isomorphism between PSGU2(q 2) and PGL2(q); Dickson refers to PSL2(q) and PGL2(q) rather than PSU2(q 2) and PSGU2(q2). In the exceptional case,

ON S U B G R O U P S OF THE S P E C I A L L I N E A R G R O U P 299

PSL2(3), PGL2(3) and PSL2(9 ) are isomorphic to A4, S 4 and A6, respec- tively. There are three proper subgroups of A 6 properly containing A 4 and each is a maximal subgroup of/16 containing A 4 as a maximal subgroup; S 4 is one of these, and the other two are conjugate copies ofA 5 (a geometri- cal explanation of this is given in [5]).

RESULT 2 (Mitchell [10]; Hartley [6]). I f K = GF(q2), then any proper subgroup of SL3(q 2) containin9 SU3(q 1) lies in SGUa(qZ).

Mitchell and Hartley actually show that PSU3(q 1) is maximal in PSL3(q2); the result as stated follows from the fact that PSU3(q 2) = PSGU3(q 2) and SL3(q 2) is perfect.

A transvection in SL,(K) is a map of the form

: z ~ z + p(z) • x,

where x is a non-zero vector in V and p is a linear form on V with p(x) = 0; it is said to be centred on x and to have axis p-l(0). For each pair of subspaces P c_ H of dimension 1 and n - 1, respectively, the commutative subgroup of SL,(K) generated by all transvections with x ~ P and p-1(0) = H will be denoted by X(P, H). Note that p is given by p(z) = (v, z) for some v e (x) ' . A transvection of the form given above will lie in SU,(K) if and only if x is isotropic and p(z) =/2(x, z) for some/2 in the multiplica- tive group K* of K such that/~ = -/2.

Recall that the general unitary group is defined by G U , ( K ) =

{9 ~ GL,(K): (gx, 9Y) = 2o(x, Y) Vx, y ~ V} where 20 ~ K* is dependent on 9 and is called the multiplicator of 9- Given that

'~o(Y, x) = (gy, gx) = (gx, gy) = £o(x, y),

the set of all )~o is a subgroup M of the multiplicative group K~ of K o; M is called the multiplicator group of C.

P R O P O S I T I O N 1. GU,(K) is the stabilizer in GL,(K) of the set of isotropic vectors of V.

Proof Clearly GU,(K) stabilizes the set of isotropic vectors. Suppose that 9 ~ GL,(K) lies in the stabilizer and let a and b be isotropic vectors such that (a, b) = 1; then 9(a) and 9(b) are isotropic and as (9(a), 9(b)) cannot be totally isotropic, (9(a), 9(b)) = 2 for some 2 e K*. Without loss of generality 9 may be replaced by any element of the double coset U,(K) 9 U,(K). Thus by virtue of Witt's theorem (cf. [1, p. 71]) we may assume that 9(a)= a and 9(b)= 2b. Let /2 e K* such that /~ = - /2 (such exists); then a +/2b is isotropic so 9(a +/2b) = a + 2/2b is isotropic and therefore 2 e K~. Let c e (a, b)', then as (c, a) and (c, b) are either both

300 O L I V E R K I N G

totally isotropie or each contain only one isotropic 1-dimensional subspace, the same must be true of (g(c), a) and (g(c), b), so g(c) ~ (a, b)'. Now let d ~ (a, b) such that c + d is isotropic; then (g(d), g(d)) = 2(d, d) and g(c + d) is isotropic, so (g(c), g(c))= 2(c, c). Thus (9(0, g(v))= 2(v, v) for all v ~ V. Finally, for any u, v ~ V, a c K,

yields

(g(u) + ~g(v), g(u) + ~g(v)) = 2(u + ~v, u + ~v)

~(o(u), g(v))+&(o(v), o(u))=2~(u, v) +2&(v, u)

for all ~ ~ K. In particular,

(g(u), g(v))+(g(v), g(u))=A(u, v)+2(v, u),

SO

&(g(u), g(v)) + &(g(v), g(u)) = )~&(u, v) + 2&(v, u),

and therefore

(~ - ~ ) ( g ( u ) , g ( v ) ) = 2(& - ~ ) (u , v)

for all ~ e K. Hence (9(u), g(v)) = 2(u, v) and g e GU,(K) with multiplicator 2. []

Note. For v ~> 2, the above result is proved by Tits in [13, Th. 8.6].

P R OP OS ITI ON 2. GUn(K ) is self-normalizing in GLn(K ) and is the nor- realizer in GLn(K ) of SUn(K).

Proof Let g ~ GL,(K) normalize SU,(K); then for any non-zero isotropic vector x ~ V, there is a unitary transvection v with centre x. The vector g(x) is the centre of the unitary transvection g~g-1 and is therefore isotropic. Thus g stabilizes the set of isotropie vectors and hence by Proposition 1,

g ~ GUn(K ). Any transvection in GU,(K) lies in SUn(K), and SUn(K) is generated by

its transvections, so any element of GLn(K ) normalizing GUn(K ) also nor- malizes SUn(K). Hence GUn(K ) is the normalizer in GLn(K ) of both GUn(K)

and SUn(K ). []

Let us now write G = SGUn(K) and Go = SUn(K) and let F ~< SLn(K ) such that Go ~< F but F ~ G; we show that F = SLn(K ). As G does not act transitively on the 1-dimensional subspaces of V, it is clear that G < SLn(K ).

O N S U B G R O U P S O F T H E S P E C I A L L I N E A R G R O U P 301

2. THE CASE n>~3

We exclude throughout this section the case n = 3 and K = GF(4) which

would require separate consideration and which is anyway dealt with by

Result 2.

P R O P O S I T I O N 3. I f n >>,3 and if K ~ GF(4) when n = 3 , then F = SL,(K).

Proof Let h ~ F~(F ~ G), then by Proposition 1 there is an isotropic

vector x ~ V such that h(x) is non-isotropic. Let Zo be a transvection in G o

with centre x, then for some v ~ (h(x))' , h% h- 1 is given by

h% h- 1 : z ~-+ z + (v, z)h(x);

it is a transvection with centre h(x) and axis ( v ) ' = h((x)'). As (h(x)) ' is

non-isotropic of dimension >~ 2 it is acted on irreducibly by SU((h(x))'), so there exists 9 E SU((h(x)) ') such that 9(@)) ~ (v) . Hence F contains the

transvection h-19hzo h-19-1h which has centre x and axis h-19( (v ) ') ( x ) ' ; let this transvection be denoted by z and let w ~ ( x ) ' so that z is

given by

~: z ~ z + (w, z)x,

with w ¢ ( x ) because the axis ( w ) ' of z is not (x ) ' . Observe that (w) ' contains an isotropic vector y such that (x, y) = 1,

because either w is non-isotropic in which case (w) ' is non-isotropic and

contains x so that y exists as claimed (using arguments given for example in

[3]), or w is isotropic and any complement of (w) in ( w ) ' containing x is non-isotropic (cf. [-8, Prop. 2.12]) and therefore contains a suitable vector y.

Thus w e (x, y) ' . We now show that F contains every transvection with

centre x and axis (w) ' . For c~ ~ K* define a transvection G by

and let

~ : z ~ z + ~(w, z)x

K, = { a ~ K * : r ~ E F } • {0};

then K~ is an additive subgroup of K containing 1. For any ~ E K~, let

g¢ ~ SU((x, y)) be the map: x~-*~x, y~._~-ly; then geGg~ 1 = z¢~ so that K o _~ K~ and K 1 may be considered as a vector subspace of K when K is thought of as a 2-dimensional vector space over Ko. We consider two possibilities. First suppose that w is isotropic and let u be an isotropic

vector in (x, y ) ' such that (w, u) = 1. Let t /~ K* such that t/ - ~-1 ~ Ko (given any f l ~ K \ K o, one of fl, f l + 1 will give such a q) and let

302 O L I V E R K I N G

~0 ~ SU((x , y, w, u>) be the m a p : x~-~qx, y~--+~l-ly, w~-~rl lw, u~--~flu; then

.0~0 i = z,q-1 e F so K o =p K I and therefore K1 = K. The second possi- bility is that w is non-isotropic. Except when K = GF(4), there exists ~ e K*

such that ( 2 ( -1 ¢ Ko (cf. [8, Cot. to Prop. 2.15]). Let ~ s SU((x , y, w)) be the m a p : x~--,~x, y~._~(-ly, w~._~(~-lw; then ~ - 1 = z~2~-~ e F, so again

K 1 = K. If K = GF(4) (so n >/4) and if w is non-isotropic, then there is a

non-isotropic vector u e (x , y, w) ' ; let co s K\Ko and let g* ~ SU((x, y, u>) be the m a p : x~--~cox, y~--~d)-ly, u~--~6)co lu; then g * r g * - i = % s F, so once more K 1 = K. Hence in each case K 1 = K, i.e. F contains X((x), (w>').

As SU((x , y>') acts irreducibly on (x , y>', there are elements gl, g2 . . . . .

g . - 2 ~ SU((x , y>') such that if wl = gi(w), then {wl, w2 . . . . . w._2} is a base for (x , y>'; there is moreove r an element g . - 1 ~ Go such that g ._ l (x) = x

and g . - l ( w ) ~ (x , y>', say g. x(W)= w._ l (for example, g . - , could be a

suitably chosen semi-transvect ion centred on x: cf. 18] for definitions). Thus

{wl, w2 . . . . . w.-1} is a base for (x>'. For each i, F contains g iX( (x> , (w>')gF 1 = X((x>, (w~>') and therefore contains the t ransvect ion

z z + z ) x

for all 2z e K; F then also contains the t ransvect ion

Y X,(wl, z ) x = z + , iw ,z x i = l i = 1 i

for all 21, 2z, . . . , 2,_ 1 1~ K. Hence F contains every transvect ion centred on

X.

To conclude that F contains every t ransvect ion in SL,(K) and is therefore the whole of SL,(K) we need only show that F acts transitively on the

1-dimensional subspaces of V, s incefX( < x > , H)f-1 = X(( f (x)) , fH) for any f E F and for any (n - 1)-dimensional subspace H containing x. As a consequence of Witt 's theorem the orbits of G O acting on the 1-dimensional subspaces of V are given by: ( a ) and ( b ) are in the same orbit if and only if (a, a) = 2,f(b, b) for some 2 ~ K*. N o w (x, y ) contains representatives of each orbit of Go, and X((x), (x)')<~ F acts transitively on the 1- dimensional subspaces other than ( x ) of (x , y) , so F must act transitively on the 1-dimensional subspaces of V. Hence F = SL,(K). [ ]

ON S U B G R O U P S OF T H E S P E C I A L L I N E A R G R O U P 303

3. T H E C A S E n = 2

We shall assume throughout this section that n = 2 and K ¢ GF(4), GF(9) or GF(25). With some amendment, the methods of proof used in this section would be valid for K = GF(4) or GF(25), but those cases are dealt with by Result 1 anyway.

The isomorphism between SU2(K) and SL2(Ko) referred to in the intro- duction may be given as follows. If x and y are isotropic vectors such that (x, y) = 1 and if # e K* such that /] = - # , then with respect to the base

{x, #y}, SUz(K) is represented by the matrices of SL2(K) with coefficients in Ko. We shall assume that Go is so represented. Given that Go acts doubly transitively on the isotropic 1-dimensional subspaces and that G preserves the set of isotropic 1-dimensional subspaces (Proposition 1), an arbitrary element of G may be multiplied by suitable elements of Go to give a diago- nal matrix

for some f l e K*; such a matrix normalizes G o if and only if f12 e K~. An immediate conclusion is that since K is separable over K o, G must be the same as Go when K has characteristic 2. Let

where 2 E K*; then g~ e G o and h~ normalizes G o and G.

P R O P O S I T I O N 4. I f K ~ GF(4), GF(9) or GF(25), then F contains a

transvection centred on x that does not lie in G.

Proof. L e t f e F\(F ~ G) and suppose t ha t f i s given by

where e, fl, 7, 6 ~ K. Note that pre-multiplication [resp., post- multiplication] by the transvections

of Go (where ~/e Ko) represents adding a multiple of one row [resp., column] to another.

304 O L I V E R K I N G

If ~ and 7 are linearly dependent over Ko, then by pre-mul t ip ly ingfby a suitable element of Go we may t r ans fo rmf in to a matrix of the form

(: :0=(; :0( 0 As this element does not lie in G, either ~z ¢ Ko or ~ lfl 6 Ko (or both). But now if 2 ~ K* with 2 2 :# 1, then F contains both

and

and

with h being dependent only on 7.

which may

one of which must be a transvection centred on x but not lying in G.

Alternatively suppose that ~ and 7 are linearly independent over Ko; then

by pre-multiplying f by suitable elements of G o we obtain an element of F~(F n G) with 1 in the (1, 1) position, i.e. we may assume that ~ = 1 and

7 ¢ Ko. Moreover, by adding K o multiples of row 1 to row 2 we may replace ~ by ~ + x for any x ~ K o and thereby assume that if ~2 = e~ +

where e, ~ ~ K o (necessarily with ~ @ 0 and, when K has characteristic 2, e@0), then ~ @ - 1 , - 2 , - 3 (because ( y + x ) z = ( e + 2 x ) ( 7 + x ) + ~ - ~/£ _ _ / ( 2 and as K o @ GF(2), GF(3), or GF(5), there exists ~ E K o such that

- - e /£ - - /£2 ~ - - 1 , - - 2 , - - 3). We now have

1:,,) for some fl e K, so F contains

- 7 2 1 + 7 - e y - ~ . 1 + 7

be transformed successively into

1 + 7 ET+ ' ( e - - 1 ) ~ + ~ + l '

ON SUBGROUPS OF THE SPECIAL LINEAR GROUP 305

Now

h2 = (1 -t- •2 - 3yg ( 7 - e ) ( ¢ + 2 ) ) ~,(¢+2) 7 z - ~ e + ¢ 2 + 2 ¢ + 1

= ( ¢ + 1 (7 - e)(¢ + 21) ) 7(¢+2) ¢ 2 + 3 ¢ + '

so F contains

( 1 (7(-g)(¢+ 2)(¢+ 1) -~'] 9¢+~h2= Y(4+2)(¢+ 1) ¢ 2 + 3 ¢ + 1)(¢+ 1) J

and if 2 = (4 + 2)(4 + 1), then h a Fh~ 1 contains

hzg~+i lh2h~l=( i (¢2(~-e)(¢+2)2+3¢+ 1)(¢+ 1 ) ) = h ' s a y '

As the construction of h from fuses only elements of G o together with f, as h is dependent only on 7 and as Go <~ ha Fh:~ 1, we may repeat the argu- ments above with h* in place off and h a Fh~ ~ in place of F, concluding that h E h a Fh E 1. Thus h a Fh[ 1 contains

h-ih* = ( ¢ + 1 - ( ~ - e)) • ( i (7 - e)(¢ + 2) 2 ) - y 1 (C a 4- 3¢ 4- 1)(¢ 4- 1)

= ( I 0 (¢+ 1)(7-g)(¢+3))1

and F contains

h~lh l h*ha=(~ (Y-g) (¢+3)(4+2)- l ) l

which is a transvection centred on x not lying in G. []

PROPOSITION 5. I f K ~ GF(4), GF(9) or GF(25), then F = SLz(K ). Proof. For ~ 6 K*, let

and let

then K1 is an additive subgroup of K containing Ko, and by Proposition 4, K1 6 Ko. Let 0 ~ KI\Ko; then for any 2 ~ K*, F contains ga~ogf 1 = ~2o.

306 OLIVER KING

If K does not have characteristic 2, then every element of K o may

be expressed as the difference between two squares, so that F contains za0

for all fl ~ K o and therefore K1 contains fl0 + c~ for all ct, fl ~ Ko, i.e. K1 = K. If K has characteristic 2, then 0 2 = e0 + ~ for some e, ¢ ~ K~ (e :p 0

because K is separable over Ko). Let 4)= 0 + ~e-1; then 4 ) - 1 = ~2~-20 + e(,~ 2 "F ~)~-2 SO K 1 contains both 4) and 4)-1.

Thus F contains

(~ 10)'(10 4);1)'(01 ~)=(4)1-1 01)

and

(; 0)(; ;)(0 and therefore

(0 for all ~ e Ko, i.e. 4)2(0 + ct) e K1 for all c~ e Ko.

Now

4)2(0 + ~) = (0 2 + ~2~-2)(0 + c0

so /~0 e K 1 for all /~ E K o . Hence K 1 = K. It follows that, whatever the

characteristic of K, F contains every transvection in SL2(K ) centred on x.

In consequence, Stabvx acts transitively on the 1-dimensional subspaces other than (x ) , so F acts transitively on the 1-dimensional subspaces of V.

As f X ( ( x ) , ( x ) ) f - 1 = X ( ( f ( x ) ) , ( f ( x ) ) ) , F contains every transvection in

SL2(K ) and is therefore the whole of SL2(K ). []

Propositions 3, 5 together with Results 1, 2 give Theorem I.

4. RELATED RESULTS Let M be the multiplicator group of C and let M1 be the subgroup of M consisting of the multiplicators of elements of SGU,(K).

C O R O L L A R Y TO T H E O R E M I. SGU,(K) is maximal in SLn(K), and except when n = 2 and K = GF(9) the proper subgroups of SL.(K) containing

SU.(K) are in one-to-one correspondence with the subgroups of M 1.

ON S U B G R O U P S OF T H E S P E C I A L L I N E A R G R O U P 307

Proof The maximal i ty of SGUn(K ) is immedia te f rom Theorem I.

Let 0: S G U n ( K ) ~ M~ be the m a p taking an element to its mul t ip l icator ; then 0 is an ep imorph i sm with kernel SUn(K ). Thus there is a one- to-one

cor respondence between the subgroups of SGUn(K ) containing SUn(K ) and the subgroups of Mr , and the result follows f rom Theorem I.

We now consider the groups GUn(K), PGUn(K), PSGUn(K), SUn(K ) and

PSUn(K) and determine condit ions for them to be maximal in GLn(K), PGLn(K), PSL,(K), SLn(K ) and PSL,(K) respectively. We then interpret these condit ions when K = C and K o = ~, and when K is finite.

Let M o = {2,~: 2 ~ K*}; then M o ~< M because the centre Z of GLn(K ) lies inside G U , ( K ) and the element of Z taking x to 2x for all x ~ V has

mul t ip l icator 2~(, and M ~< K* (cf. Section 1). If n is odd, then G U , ( K ) =

Z • U,(K) (cf. [-3, p. 77]) so M o = M. If 9 e G U , ( K ) has mult ipl icator a and de te rminant A, then when n is odd A = 4"# where 4 ( = a and #~ = 1 and when n is even, say n = 2m, then A = am# where #/~ = 1 (cf. [-3, p. 77]); in

either case AA = an. Conversely, if A ~ K* such that AA = a n for some a ~ M, then when n is odd AA = (~()n = ~n~n for some ~ ~ K* so A = 4"#

when # = A~-" and #~ = 1, and when n is even A = am# when # = Aa -m

and #/Tt = 1. As Un(K) contains elements of de terminant # for any # ~ K* with #~ = 1 (e.g. a m a p taking a non- isot ropic vector v to #v and fixing

each vector in (v ) ' , i.e. a quasi-symmetry) , it follows that given A ~ K*,

there is an element of GUn(K) with de te rminant A if and only if AA = ~" for some ~ ~ M.

T H E O R E M II. GUn(K ) is maximal in GLn(K ) if and only if every element of M o has an nth root in M.

Proof Let J = {A ~ K*: det 9 = A for some # ~ GUn(K)}. If J @ K*, then G U , ( K ) is proper ly conta ined in the p roper subgroup of GL, (K) con-

sisting of those elements with de te rminant in J. On the other hand, if

J = K* and if G U , ( K ) < F ~< GL, (K) then SGUn(K ) < F n SLn(K) so by the Corol la ry to Theorem I, SLn(K ) ~< F whence F = GL, (K) and therefore

GUn(K ) is maximal in GL,(K). Thus G U , ( K ) is maximal in GLn(K ) if and only if J = K*. N o w J = K* if and only if for any A ~ K* there exists

c~ ~ M such that A,~ = a n, i.e. if and only if every element of M o has an nth root in M. [ ]

Given tha t Z ~< GU,(K) , as an immedia te consequence of Theorems I and I I we have

T H E O R E M III . P G U , ( K ) is maximal in PGLn(K ) /f and only if every element of M o has nth root in M, and PSGUn(K) is maximal in PSL,(K). [ ]

3 0 8 O L I V E R K I N G

Clearly SU.(K) is maximal in SL.(K) if and only if SU.(K) = SGU.(K), i.e. if and only if every element of GU.(K) with determinant 1 has multi-

plicator 1. If SGU.(K) contains an element with multiplicator a 4: 1, then

1 • i = ~", i.e. M contains a non-trivial nth root of 1. Conversely, if ~ ~ M is a non-trivial nth root of 1 and if g ~ GU.(K) has mul t ip l i ca to r . , with

det g = A say, then AA = ~" = 1, so multiplying g by an element of U.(K) with determinant A- ~ gives an element of SGU.(K) with multiplicator ~. This proves

T H E O R E M IV. SU.(K) is maximal in SL.(K) if and only if M contains no non-trivial nth roots of 1. []

By the Corollary to Theorem I, the question of when PSU.(K) is

maximal in PSL.(K) is equivalent to the question of when Z~ - SU.(K) =

SGU.(K) where Z~ = Z c~ SL.(K) is the centre of SL.(K). Let M i and m 2

be the subgroups of M consisting of the multiplicators of elements of SGU.(K) and Z~ - SU.(K) respectively. Clearly if Zx • SU.(K) = SGU.(K),

then M 1 = m 2 . On the other hand, if M~ = M 2 , then given g ~ SGU.(K) with multiplicator ~, there exists h ~ Z~ • SU.(K) with multiplicator ~; the multiplicator of h - Xg is 1, i.e. h - lg ~ SU.(K), so g ~ Z~ - SU.(K) and there-

fore SGU.(K) = Z~ • SU.(K). Thus SGU.(K) = Z 1 - SU.(K) if and only if

M~ = M 2 . N o w m z consists of the elements of K~ of the form ~ where ~ K* such that 4" = 1, and as shown above, M~ consists of the elements

~ M such that ~" = 1. We therefore deduce

T H E O R E M V. PSU.(K) is maximal in PSL.(K) if and only if every nth root of 1 lying in M may be written ~(for some nth root ~ of 1 in K*. []

Suppose that K = C and that for 2 ~ K, 2 is the usual complex conju-

gate; then K o = ~. Every element 22 of K o is positive and every positive

element of K o may be written in the form 2X (because every positive

element of K o is a square in Ko), so M o is the set of positive real numbers. If n is odd, then M = M o; if n is even and n = 2v, then M = K*. If n is even and 2v < n, then there are totally isotropic v-dimensional subspaces U and W of V such that U c~ W = {0}; the subspace 17o = (U + W)' is then aniso- tropic (and non-isotropic) and if Co is the restriction of C to V 0 , then M is

the multiplicator group of Co (cf. [3, pp. 23, 77]). As Co must take only positive values or only negative values on Vo\{0}, M = Mo. Thus M = Mo whenever v < n/2. Theorems II, IV, V together give

O N S U B G R O U P S OF T H E S P E C I A L L I N E A R G R O U P 309

T H E O R E M VI. Let K = C and let f~ be the usual complex conjugate of 2

for all 2 ~ K. Then (i) G U , ( K ) is maximal in GL,(K);

(ii) SU,(K) is maximal in SL,(K) i f and only i f v < n/2;

(iii) PSU,(K) is maximal in PSL,(K) i f and only if v < n/2. [ ]

N o w suppose that K = GF(q2); then K o = GF(q) and 2 = 2 q for all

2 E K. Let 0 be a primitive element of K (i.e. 0 is a genera tor of the multiplicative group K*); then K~ is generated as a multiplicative g roup by

0 q+l (= 00), SO m o = M = K~.

T H E O R E M VII. Let K = GF(q2). Then

(i) G U , ( K ) is maximal in GL,(K) i f and only if(q - 1, n) = 1 ;

(ii) SU,(K) is maximal in SL,(K) i f and on ly / f (q - 1, n) = 1;

(iii) PSU, (K) is maximal in P S L , ( K ) / f and only if either n is odd or if n is even and (qz _ 1)/(q2 _ 1, n) is odd.

Proof For any posit ive integer k, the m a p Wk: K ~ - - ~ K * , 2~- .2 k is a

g roup h o m o m o r p h i s m and is a bijection if and only if (q - 1, k) = 1. Every

element of K* has an nth root in K~ if and only if ud, is surjective, and K* contains no non-trivial nth roots of 1 if and only if ud, is injective. As the

terms 'b i jec t ive ' , ' su r j ec t ive ' and ' in jec t ive ' are equivalent for maps f rom

Ko* to K * , (i) and (ii) follow from Theorems II, IV.

Let d = (q - 1, n) and let e = (q - 1)/d; then the nth roots of 1 in K * ( = M) are the elements 0 (q+ l)et for t = 1, 2 . . . . , d. If O (q+ l)e = ~ with ~ e K*

such that ~ " = 1, then Otq+l)et= ~(()~ and ~ " = 1, so by Theorem V, PSU,(K) is maximal in PSL,(K) if and only if we can write 0 (q+ 1)e = ~ ( with 4" = 1. N o w if ~ ( = 0 tq+l)e, then ~ = 0 e+s(q-1) for some s e {0, 1 . . . . . q}, so

4 " = 0"e+"s(q-l) = 1 if and only if q 2 l i n e + n s ( q - 1), i.e. if and only if

q + 1 [n(1 + sd)/d. Thus PSU, (K) is maximal in PSL,(K) if and only if there exists s ~ {0, 1, . . . , q} such that q + 1 In(1 + sd)/d. If n is even and (q2 _ 1)/ (q2 _ 1, n) is even, then q - 1 is even, so d is even, 1 + sd is odd for all s

and q + 1 is divisible by a higher power of 2 than n/d, whence q + 1Xn(1 + sd/d) for any s. Let 1 = (q + 1)/(q + 1, n/d) = d(q + 1)/(q 2 - - 1, n). If n is odd

or if n is even and ( q 2 _ 1 ) / ( q 2 1, n) is odd, then (l, d ) = 1, because ( l , d ) ~ < ( q + 1, q - 1 ) = 1 or 2, d is odd if n is odd, and l is odd if (q2 - 1)/ (q2 _ 1, n) is odd. Therefore there exist integers r, t such that rl + td = 1, i.e. l [ 1 - td for some integer t. By adding a suitable mult iple of (q + 1)d to 1 - td we see that l] 1 + sd for some s ~ {0, 1, . . . , q}, so q + 1 In(1 + sd)/d.

This proves (iii). [ ]

310 O L I V E R K I N G

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A u t h o r ' s address :

O l i v e r King ,

S c h o o l of M a t h e m a t i c s ,

T h e Un ive r s i t y ,

N e w c a s t l e upon T y n e ,

N E 1 7 R U ,

E n g l a n d

(Received, November 8, 1984)