EXOTIC NORMAL FUSION SUBSYSTEMS OF GENERAL LINEAR

                                     GROUPS



                                    ALBERT RUIZ


       Abstract. We classify the saturated fusion subsystems of index prime to *
 *p of the general
       linear group over Fq over a Sylow p-subgroup, where q is a prime power p*
 *rime to an odd
       prime p. In this classification we get some of the exotic p-local finite*
 * groups discovered by
       C. Broto and J. Moller as saturated fusion subsystems of the general lin*
 *ear group.
                                 1. Introduction


  The concept of p-local finite group arose in the work of C. Broto, R. Levi an*
 *d B. Oliver [3]

as a formalization of the p-local structure of a finite group. A p-local finite*
 * group consists of

a triple (S, F, L) where S is a p-group, F is a category defined in an axiomati*
 *c way which

models the fusion category over S, a Sylow p-subgroup of a finite group, and L *
 *is an extension

of F which contains extra information so that its p-completed nerve has many of*
 * the same

properties as the p-completion of the classifying space of a finite group.

  One source of examples of p-local finite groups is the ones constructed from *
 *finite groups:

when G is a finite group and S is a Sylow p-subgroup in G, we can construct a t*
 *riple

(S, FS(G), LcS(G)) which is a p-local finite group.  We can recover the p-prima*
 *ry informa-

tion of G using the fact that in this case |BG|^p' |LcS(G)|^p.

  The examples which cannot be constructed from a finite group are called exoti*
 *c examples.

There are known exotic examples of p-local finite groups for all primes p, and *
 *these have been

constructed in two different ways:

  (i)Examples by C. Broto, R. Levi and B. Oliver ([3], [4]), and A. D'iaz, A. V*
 *iruel and the

     author ([6], [10]) are constructed in a combinatorial way: they start with*
 * the saturated

     fusion system of a finite group and they add morphisms to the automorphism*
 * group of

     some proper subgroups.

     Also Solomon's example by R. Levi and B. Oliver ([7]): they show that the *
 *fusion on the

     Sylow 2-subgroup of Spin7(q) (q an odd prime power) considered by Solomon,*
 * which is

     known not to occur as the fusion of any finite group, has a p-local finite*
 * group structure.

 (ii)Homotopy fixed point sets of p-compact groups by C. Broto and J. Moller [5*
 *].

  In this paper we use the study of saturated fusion subsystems by C. Broto, N.*
 * Castellana,

J. Grodal, R. Levi and B. Oliver [2] to obtain some of the classifying spaces o*
 *f exotic p-local

finite groups from [3] and [5] as, up to homotopy equivalence, covering spaces *
 *of the classifying

spaces of a non-exotic p-local finite groups (Remark 6.4).

   ______________
   Key words: 2000 Mathematics subject classification 55R35, 20D20.
  Partially supported by FEDER-MEC grant MTM2004-06686.
                                         1


2                                   ALBERT RUIZ


  More exactly we classify the saturated fusion subsystems of index prime to p *
 *of the general

linear group GL n(q) for q a prime power prime to p.

  Now we will describe the contents of the paper, giving simplified versions of*
 * the main

results.

  In Section 3 we review the classification of the saturated fusion subsystems *
 *of index prime

to p from [2]. For any (S, F, L) a p-local finite group, this classification is*
 * given in terms of
                                   0
the F-centric subgroups of S and Op*(F), the smallest subcategory with the same*
 * objects as

F and which contains all the restrictions of all the automorphisms in F of p-po*
 *wer order.

With this data there is a bijection between saturated fusion subsystems of inde*
 *x prime to p

in F and subgroups of


                             p0(F) def=OutF(S)= Out0F(S) ,


where Out 0F(S) is defined as


          Out0F(S) def=<ff 2 OutF (S) | ff|P 2 Mor Op0*(F)(P, S), some F-centri*
 *cP > .


                          0
  If we want to compute Op*(F) in some particular cases we would like to restri*
 *ct the family

of subgroups of S involved in this computation to a smaller one. So in this sec*
 *tion we give

the following general result (Theorem 3.4):


Theorem A. Let F be a saturated fusion system over S.  Then for each morphism _*
 * 2

Hom Op0*(F)(P, P 0), there exists a sequence of subgroups of S


                    P = P0, P1, . .,.Pk = P 0 and  Q1, Q2, . .,.Qk,


and morphisms _i2 AutOp0*(F)(Qi), such that

     o Qi is fully F-normalized, F-centric and F-radical for each i;

     o Pi-1, Pi  Qi and _i(Pi-1) = Pi for each i; and

     o _(u) = _k O _k-1O . .O._1(u), 8u 2 P .


  In Sections 4 and 5 we use J.L. Alperin and P. Fong results in [1] to describ*
 *e the possible F-

centric, F-radical subgroups in GL n(q). This description enables us to classif*
 *y the saturated

fusion subsystems of index prime to p in GL n(q), using the following result ob*
 *tained here as

Theorem 5.10:


Theorem B. Let p be a prime and q a prime power prime to p. Let e be the multip*
 *licative

order of q modulo p, and n   ep. Consider (Sn,q, Fn,q, Ln,q) the p-local finite*
 * group induced

by GL n(q) over Sn,q, a Sylow p-subgroup. Then


                                   p0(Fn,q) ~=Z=e .


  In Section 6 we identify the p-local finite groups corresponding to these sat*
 *urated fusion

subsystems, getting that these correspond to the p-local finite groups describe*
 *d by C. Broto

and J. Moller as the finite Chevalley version of the generalized Grassmannians,*
 * denoted by

X(e, r, m)(q0), for any positive integers e, r, m such that r divides e, e divi*
 *des (p - 1) and q0

a p-adic unit. More precisely the result that we get in Proposition 6.1 and The*
 *orem 6.3 is:


            EXOTIC NORMAL FUSION SUBSYSTEMS OF GENERAL LINEAR GROUPS           3


Theorem C. Fix p, q, e and (Sn,q, Fn,q, Ln,q) as in Theorem B, and, for each r *
 *a divisor of

e, let (Sn,q, Fn,q,r, Ln,q,r) the unique p-local finite group such that (Sn,q, *
 *Fn,q,r) is the saturated

fusion subsystem of index prime to p in (Sn,q, Fn,q) such that Out Fn,q(Sn,q)= *
 *OutFn,q,r(Sn,q) ~=

Z=r. Then:

(a) Up to homotopy equivalence, there is a fibration


                             |Ln,q,r| ! |Ln,q| ! B(Z=r) .


(b) |Ln,q,r| ' BX(e, r, [n=e])(qe) up to p-completion, where [n=e] is the great*
 *est integer less

    or equal than n=e.


  Using that BX(e, r, m)(q0) are known to be exotic p-local finite groups when *
 *m   p and

r > 2 [5, Proposition 11.5], this result provides us examples of extensions of *
 *p-local finite

groups where one of the involved elements is exotic, and the other two correspo*
 *nd to finite

groups. These examples answer a question proposed by B. Oliver in the Banff con*
 *ference on

Homotopy theory and group actions (November 2005).


Remark 1.1. These results give us a new construction of the generalized p-adic *
 *Grassman-

nians. For all integer j   0 we have natural maps |Ln,qpj,r| ! |Ln,qpj+1,r| tha*
 *t at the level of

maximal tori induce the inclusion


                            (Z=pl+j)[n=e]  (Z=pl+j+1)[n=e],


where l =  p(qe - 1). The telescope construction of these maps gives us a homot*
 *opy equiva-

lence

                           BX(e, r, [n=e]) ' hocolim|L   pj|
                                                j     n,q ,r

up to p-completion.


Acknowledgements. The author would like to thank to Carles Broto and Antonio Vi*
 *ruel

the many helpful discussions during our stay at the Centre Interfacultaire Bern*
 *oulli, where

this project started. The author is also grateful to Bob Oliver for the helping*
 * discussion when

proving last results in this paper.

  The author would also like to thank to the Centre Interfacultaire Bernoulli (*
 *Lausanne,

Switzerland) and the Institut Mittag-Leffler (Djursholm, Sweden) for their supp*
 *ort and hos-

pitality while working on this paper.

  The author also thanks the referee for the improvements suggested.


                             2. p-local finite groups


  In this section we review the concept of a p-local finite group introduced in*
 * [3] that is based

on a previous unpublished work of L. Puig, where the axioms for fusion systems *
 *are already

established. See [4] for a survey on this subject.

  If P and Q are subgroups of a group G we consider Hom G (P, Q) the group morp*
 *hisms

from P to Q induced by conjugation of elements in G, and Inj(P, Q) are the inje*
 *ctive group

morphisms from P to Q.


Definition 2.1. A fusion system F over a finite p-group S is a category whose o*
 *bjects are the

subgroups of S, and whose morphisms sets Hom F (P, Q) satisfy the following two*
 * conditions:


4                                   ALBERT RUIZ


(a) Hom S (P, Q)   Hom F(P, Q)   Inj(P, Q) for all P and Q subgroups of S.

(b) Every morphism in F factors as an isomorphism in F followed by an inclusion.


  We say that two subgroups P ,Q   S are F-conjugate if there is an isomorphism*
 * between

them in F. As all the morphisms are injective by condition (b), we write by Aut*
 * F(P ) the

group Hom F(P, P ). Out F(P ) denotes the quotient group AutF (P )= AutP(P ).

  The fusion systems that we consider are saturated, so we need the following d*
 *efinitions:


Definition 2.2. Let F be a fusion system over a p-group S.

     o A subgroup P   S is fully F-centralized if |CS(P )|   |CS(P 0)| for all *
 *P 0which is

       F-conjugate to P .

     o A subgroup P   S is fully F-normalized if |NS(P )|   |NS(P 0)| for all P*
 * 0which is

       F-conjugate to P .

     o F is a saturated fusion system if the following two conditions hold:

        (I)Every fully F-normalized subgroup P   S is fully F-centralized and A*
 *ut S(P ) 2

           Sylp(Aut F(P )).

       (II)If P   S and ' 2 Hom F(P, S) are such that 'P is fully F-centralized*
 *, and if we

           set

                       N' = {g 2 NS(P ) | 'cg'-1 2 AutS('P )},


           then there is __'2 Hom F(N', S) such that __'|P = '.


  As expected, every finite group G gives rise to a saturated fusion system [3,*
 * Proposition

1.3], which provides valuable information about BG^p. Some classical results fo*
 *r finite groups

can be generalized to saturated fusion systems, as for example, Alperin's fusio*
 *n theorem for

saturated fusion systems [3, Theorem A.10]:


Definition 2.3. Let F be any fusion system over a p-group S. A subgroup P   S i*
 *s:

     o F-centric if P and all its F-conjugates contain their S-centralizers.

     o F-radical if Out F(P ) is p-reduced, that is, if Out F(P ) has no nontri*
 *vial normal p-

       subgroups.


Theorem 2.4 ((Alperin's fusion theorem for saturated fusion systems)). Let F be*
 * a saturated

fusion system over S. Then for each morphism _ 2 AutF (P, P 0), there exists a *
 *sequence of

subgroups of S

                    P = P0, P1, . .,.Pk = P 0 and  Q1, Q2, . .,.Qk,


and morphisms _i2 AutF (Qi), such that

     o Qi is fully F-normalized, F-centric and F-radical for each i;

     o Pi-1, Pi  Qi and _i(Pi-1) = Pi for each i; and

     o _(u) = _k O _k-1O . .O._1(u), 8u 2 P .



              3. Saturated fusion subsystems of index prime to p


  The saturated fusion subsystems of index prime to p in any saturated fusion s*
 *ystem (S, F)

are described in [2] in terms of:


            EXOTIC NORMAL FUSION SUBSYSTEMS OF GENERAL LINEAR GROUPS           5

                                                 0
Definition 3.1.      o If G is a finite group, Op (G) is the smallest normal su*
 *bgroup of

       index prime to p (equivalently the subgroup generated by elements of p-p*
 *ower order

       in G).

     o A restriction-closed category over S is a category F such that Ob (F) is*
 * the set of

       subgroups of S, such that all morphisms in F are group monomorphisms bet*
 *ween the

       subgroups, and with the following additional property: for each P 0  P  *
 * S and Q0

       Q   S, and each ' 2 Hom F(P, Q) such that '(P 0)   Q0then '|P0 2 Hom F(P*
 * 0, Q0).
                                           0
     o If F is a saturated fusion system Op*(F)   F denotes the smallest restri*
 *ction-closed
                                                      0
       subcategory of F whose morphism set contains Op (Aut F(P )) for all subg*
 *roups P   S.

     o If F is a saturated fusion system over S we write:


        Out 0F(S) = <ff 2 OutF (S) | ff|P 2 Mor Op0*(F)(P, S), some F-centricP *
 *  S> .


  The result giving the classification of the saturated fusion subsystems of in*
 *dex prime to p

is the following [2, Theorem 5.4 and Proposition 5.2]:


Theorem 3.2. For any saturated fusion system F over a p-group S, there is a bij*
 *ective

correspondence between subgroups

                                      def            0
                           H    p0(F) = Out F(S)= OutF(S)


and saturated fusion subsystems FH of F over S of index prime to p in F. The co*
 *rrespondence

is given by associating to H the fusion system generated by b`-1(B(H)), where B*
 *(H) is a

category with one object and with morphism monoid the group H, and b`is the uni*
 *que functor

                                 b`:Fc ! B( p0(F))


with the following properties:

(a) b`(ff) = ff (modulo Out0F(S)) for all ff 2 AutF (S).
                            0   c
(b) b`(') = 1 if ' 2 Mor (Op*(F) ).


  The rest of the section is dedicated to reduce the family of subgroups involv*
 *ed in the

calculation of Out 0F(S).
              0
  Consider ROp*(F) the smallest restriction-closed category of F whose morphism*
 * set con-
        0
tains Op (Aut F(P )) for all fully F-normalized, F-centric, F-radical subgroups*
 * P   S.


Lemma 3.3. Fix F is a saturated fusion system over S. Then:
                           0
(a) Aut F(S) normalizes ROp*(F).
            0
(b) F = <ROp*(F), AutF(S)>.


Proof. (a) Consider _ 2 Hom ROp0*(F)(P, P 0) and ff 2 AutF (S). We should check*
 * that ff_ff-1 2

Hom  ROp0*(F)(ff(P ), ff(P 0)).

_ is the restriction of composition of OE1, . .,.OEk, which are p-power order e*
 *lements of auto-

morphisms of F-centric, F-radical subgroups R1, . .,.Rk. But now ff_ff-1 can be*
 * written

as the composition of ffOEjff-1, which are p-power order elements of automorphi*
 *sms of ff(Rj),

which are again fully F-normalized, F-centric and F-radical because this two pr*
 *operties are

kept under Aut F(S)-conjugation.


6                                   ALBERT RUIZ


(b) By Alperin's theorem for saturated fusion systems (Theorem 2.4) it is enoug*
 *h to check
                    0
that Aut F(Q)   <ROp*(F), AutF(S)> for Q an F-centric, F-radical and fully F-no*
 *rmalized
                                                                     0
subgroup of S. We will proceed by downward induction. Aut F(S)   <ROp*(F), AutF*
 *(S)>, so
                                                                0              *
 *         0
the result holds for Q = S. Assume now that the Aut F(Q0)   <ROp*(F), AutF(S)> *
 *for all Q

of bigger order than a fixed Q, and _ 2 AutF (Q).

  Consider the subgroup K def=_ AutS(Q)_-1, which is a p-subgroup of AutF (Q), *
 *so K

Aut ROp0*(F)(Q). Since Q is fully F-normalized, AutS (Q) is a Sylow p-subgroup *
 *of Aut F(Q).

As AutR Op0*(F)(Q)   AutF (Q), we get that both Aut S(Q) and K are Sylow p-subg*
 *roups in

Aut ROp0*(F)(Q), so they are conjugated by an element OE 2 AutROp0*(F)(Q):


                                OEKOE-1   AutS(Q) .                            *
 * (1)


So,_as_F is saturated, by condition (II) in Definition 2.2 OE_ must extend to a*
 * morphism

OE_ defined over NOE_which by Equation_(1)is equal to NS(Q), so it is always bi*
 *gger than
                                                                  0
Q.  So by induction hypothesis OE_:NS(Q) ! S is an element in <ROp*(F), AutF(S)*
 *>, so
                                    0
its restriction OE_ is again in <ROp*(F), AutF(S)>.  Recall now that OE 2 Aut R*
 *Op0*(F)(Q) so
                                   0
_ = OE-1(OE_) is an element in <ROp*(F), AutF(S)>.


  Now we are ready to prove the main result of this section, which is analogous*
 * to the
                                                   0
Alperin's Theorem (Theorem 2.4) for the category Op*(F):


Theorem 3.4. Let F be a saturated fusion system over S.  Then for each morphism*
 * _ 2

Hom Op0*(F)(P, P 0), there exists a sequence of subgroups of S


                    P = P0, P1, . .,.Pk = P 0 and  Q1, Q2, . .,.Qk,


and morphisms _i2 AutOp0*(F)(Qi), such that

     o Qi is fully F-normalized, F-centric and F-radical for each i;

     o Pi-1, Pi  Qi and _i(Pi-1) = Pi for each i; and

     o _(u) = _k O _k-1O . .O._1(u), 8u 2 P .

                                   0      R p0
Proof.We have just to check that Op*(F)    O* (F).
                                      0
  Consider ff: P ! P 0an element in Op*(F). Then ff is the composition of restr*
 *ictions of

automorphisms of order a power of p in Aut F(Qi), for some subgroups Q1, . .,.Q*
 *r. So it is
                                              0
enough to check that these elements are in ROp*(F).

  Consider then ff: Q ! Q an element of order a power of p, for Q any subgroup *
 *in S. We
                                       0
will check that it is a morphism in ROp*(F) by downward induction. The result i*
 *s true for

Q = S (both morphisms sets are empty sets). Assume now that for a fixed Q, all *
 *the elements
                                             0             0
of order a power of p in Aut F(Q0) are in ROp*(F) for all Q of order bigger tha*
 *n the order of

Q.

  If Q is not fully F-normalized, consider g :Q ! Q0an isomorphism in Hom F (Q,*
 * Q0) such

that Q0 is fully F-normalized. Using Lemma 3.3 and [2, Lemma 3.4 (c)] we get th*
 *at there

exists fi 2 AutF (S) and ' 2 Hom ROp0*(F)(fi(Q), Q0) such that g = ' O fi|Q.

  If g O ff O g-1 2 Hom ROp0*(F)(Q0, Q0) then fi|Q O ff O fi-1|fi(Q)= '-1 O g O*
 * ff O g-1 O ' is also in

Hom R Op0*(F)(Q0, Q0). Use now that ff is Aut F(S)-conjugated to fi|Q O ff O fi*
 *-1|fi(Q)and Aut F(S)


            EXOTIC NORMAL FUSION SUBSYSTEMS OF GENERAL LINEAR GROUPS           7

              0                                           R p0
normalizes ROp*(F) and we obtain that ff is a morphism in  O* (F). So we can as*
 *sume that

Q is fully F-normalized.

  If Q is not F-centric, then by condition (II) in [3, Definition 1.2] we get t*
 *hat ff extends

to a morphism __'2 Hom F (Q . CS(Q), S).  The image of __'must be again in Q . *
 *CS(Q), so
' 2_____Aut              __                         __
        F(Q . CS(Q)) and ' |Q = '.  If the order of ' is a power of p, we can a*
 *pply the
induction hypothesis and the result follows.  If not, we can consider an intege*
 *r r such that
' r_____|                       __r
         Q = ' and the order of ' |Q is a power of p, and apply the induction h*
 *ypothesis.
  Assume now that Q is F-centric and fully F-normalized, and ff 2 AutF (Q) an e*
 *lement of

p-power order. If Q is F-radical, we have finished, so suppose that Q is not F-*
 *radical. That

means that Op(Out F(Q)) 6= <1>, and so K def=Op(Aut F(Q)) is strictly bigger th*
 *an AutQ (Q).

As Q is fully F-normalized, we have that AutS(Q) is a Sylow p-subgroup for Aut *
 *F(Q), and

so K   AutS(Q). Consider


                           NKS(Q) def={x 2 NS(Q) | cx 2 K} ,


which is strictly bigger than Q. If x 2 NKS(Q), ffcxff-1 is an element in K (K *
 *is a normal

subgroup in AutF (Q)), so, in this case,


                   NKS(Q)   Nffdef={x 2 NS(Q) | ffcxff-1 2 AutS(Q)}


and by condition (II) in [3, Definition 1.2] ff extends to a morphism __ff:NKS(*
 *Q) ! S. Moreover

the image of __ffis contained in NKS(Q): the extension to elements of the norma*
 *lizer in S of Q

must give elements in the normalizer of the image, which is again Q; also c_ff(*
 *x)= ff O cxO ff-1,

so if cx is in K, c_ff(x)is again in K. So we have an extension of ff, which ca*
 *n be taken of p-
                                                                  0
power order (take __ffras before if necessary) and which is in ROp*(F) by induc*
 *tion hypothesis.
                                                                        0
As we are in a restriction closed category, ff is also a morphism in ROp*(F).


         4. A representation of the extraspecial group p1+2fl+in GL n(q)


  In this section we introduce some subgroups needed to understand the fusion s*
 *ystem of

GL n(q) over a prime p such that p - q.  We divide this study in two subsection*
 *s, the first

one concerning to the particular case q   1 mod p and the second one, where we *
 *extend the

results to the general case.


Notation 4.1. Consider the extraspecial group of order p1+2fland exponent p, no*
 *ted as p1+2fl+,

as the group generated by A0, . .,.Afl-1, B0, . .,.Bfl-1and C with the relation*
 *s [Ai, Aj] = 1,

[Bi, Bj] = 1 and [Ai, Bj] = Cffiijfor all i, j 2 {0, 1, . .,.fl - 1}.


4.1. Case q   1 mod p. Fix first q a prime power such that p | (q - 1) and i a *
 *p-root of the

unity in Fq.

  We will construct a representation of p1+2fl+in GL pfl(q), so we assume that *
 *n   pfl. In the

case n > pflwe consider the inclusion of p1+2fl+in GL n(q), as the composition *
 *p1+2fl+  GL pfl

GL  n(q).

  A faithful representation can be constructed as follows: consider V  a Fq-vec*
 *tor space of

dimension pfl, with basis {v0, . .,.vpfl-1}; write each integer k such that 0  *
 * k   pfl- 1 in

base p, getting fl unique numbers ai2 {0, . .,.p - 1}:


                         k = a0 + a1p + a2p2 + . .+.afl-1pfl-1.


8                                   ALBERT RUIZ


Then, using this basis consider the following morphisms:

     o Ai is the linear transformation which sends


                           va0+...+aipi+...7! iaiva0+...+aipi+...,


     o Bi is the permutation in the basis which sends


                           va0+...+aipi+...7! va0+...+(ai+1)pi+...,


       where the coefficient (ai+ 1) is reduced mod p.


Lemma 4.2. The elements Ai and Bi defined above give a faithful representation *
 *of p1+2fl+in

GL pfl(q).


Proof.A direct computation shows us that these elements carry out the relations*
 * in Notation

4.1, taking C as i Id.


We will also be interested in the following transformation: for i 2 {0, 1, . .,*
 *.n-1} we consider

oei:V ! V as the linear map which in the basis v0, . .,.vpfl-1acts as:


                          oei(va0+...+aipi+...) def=va0+...+(,ai)pi+...,       *
 * (2)


where , is a (p - 1)-root of unity in Fp and again the coefficient (,ai-1) is r*
 *educed mod p.

  We can check with a direct computation that:
                                        (
                                          Aj    if i 6= j
                            oeiAjoe-1i=    ,-1                                 *
 * (3)
                                          Aj    if i = j

                                         ae
                                           Bj  if i 6= j
                             oeiBjoe-1i=    ,                                  *
 * (4)
                                           Bj  if i = j

        -1               ,                                    -1
where A,j (respectively Bj) means the matrix Aj to the power ,   (respectively *
 *to the power

,).


4.2. General case. Now we consider e the order of q mod p. We assume that e > 1*
 *, and

observe that e must divide (p - 1).

  Consider Fqea Galois extension of Fq, with Galois group Z=eZ. Fqecan be regar*
 *ded as the

quotient Fq[x]=r(x), where r(x) is an irreducible polynomial of degree e with c*
 *oefficients in

Fq.

  Now consider Fqeas Fq-vector space with basis {1, x, . .,.xe-1}. This gives a*
 *n inclusion of

GL 1(qe)   GL e(q), and more generally of GL n(qe)   GL en(q) (convert every co*
 *efficient of a

matrix in GL n(qe) to an e x e-matrix with coefficients in Fq).

  Using this inclusion we get that we can apply the previous subsection to the *
 *study of

GL n(qe), getting the corresponding elements Aj, Bj and oej in GL en(q), and al*
 *so the corre-

sponding inclusion of p1+2fl+  GL epfl(q).

  Consider the Frobenius automorphism of Fqe, which is the Fq-linear map which *
 *sends every

y def=a0+ . .+.ae-1xe-1 to yq, and is a generator of the Galois group of Fqeove*
 *r Fq. This can

be thought as an e x e matrix with coefficients in Fq.


            EXOTIC NORMAL FUSION SUBSYSTEMS OF GENERAL LINEAR GROUPS           9


  Consider the following fixed isomorphism Fnqe~=Fenqas Fq-vector space:

if {v1, . .,.vn} is a basis of Fnqeas Fqe-vector space, then


              {v1, xv1, . .,.xe-1v1, v2, xv2, . .,.xe-1v2, . .,.vn, xvn, . .,.x*
 *e-1vn}


is a basis of Fenqas Fq-vector space.

  Now define 'n the Fq-linear automorphism in Fnqewhich sends a vector


                 'n(r1(x)v1 + . .+.rn(x)vn) def=r1(x)qv1 + . .+.rn(x)qvn ,     *
 * (5)


and regard it as an automorphism of Fenqin the previous basis.

  With these definitions, we can compute the following conjugations:

     o 'nAj'-1n= Aqjand

     o 'nBj'-1n= Bj.

Note that, as the order of the Frobenius automorphism is qe, the inverse '-1nis*
 * given by:

                                                   e-1              qe-1
              '-1n(r1(x)v1 + . .+.rn(x)vn) = r1(x)q  v1 + . .+.rn(x)   vn ,

                             e-1                             qe-1
So '-1nsends ri(x)vito ri(x)q  vi. If we apply Aj, each ri(x)   viis multiplied*
 * by a coefficient

aijwhich only depends on the expression of i in base p and j.  In we apply now *
 *'n to
         e-1          q                                              q
aijri(x)q  vi we get aijvi. Now as Aj are diagonal matrices, we get Aj.

  Bj permutes the basis vi, so it commutes with the action of 'n.


                         5. The fusion systems of GL n(q)


Notation 5.1. In all this section fix p and odd prime and q a prime power prime*
 * to p.

Let e be the multiplicative order of q modulo p, and l =  p(qe - 1).  Consider *
 *also V  an

n-dimensional Fq-vector space, in such a way that we write GL (V ) instead of G*
 *L n(q) if we

have to deal with subgroups.


  We are interested in the F-centric, F-radical subgroups of the general linear*
 * groups. Fix

S a Sylow p-subgroup of a finite group G, and P a subgroup of S.

  Note that when G is a finite group and S a Sylow p-subgroup, then a subgroup *
 *P   S is

p-radical in G when NG(P )=P does not contain any non trivial normal p-subgroup*
 *. Also a

subgroup P   S is p-centric in G if CG(P ) = Z(P ) x C0G(P ), where C0G(P ) has*
 * order prime

to p.

  For P , being p-radical in G and being FS(G)-radical (FS(G) is the saturated *
 *fusion system

given by G over S) are independent definitions. If we require P to be p-centric*
 * in G, then

we have that when P is FS(G)-radical, P is also p-radical in G. So the list of *
 *subgroups of

G that we are interested in is contained in the list of p-radical subgroups of *
 *G.

  The possible p-radical subgroups of GL n(q) over the prime p are described in*
 * [1, Section

4] and depend on some parameters ff, fl, m, and c = (c1, . .,.ct). These are co*
 *nstructed as

follows:

  Fixed ff   0 and fl   0 consider (Z=pl+ff)p1+2fl+the central product of the c*
 *yclic group of

order pl+ffand an extraspecial group of order p1+2fland exponent p over the cen*
 *ter of the

extraspecial group. Consider Rff,flthe image of (Z=pl+ff)p1+2flby the compositi*
 *on

                                                   ff
                       (Z=pl+ff)p1+2fl+  GL pfl(qep )   GL epff+fl(q) ,        *
 * (6)


10                                  ALBERT RUIZ


where the first inclusion works as follows:

     o The subgroup Z=pl+ffcorresponds to the matrices of the form ~ Id, where *
 *~ is a

       pl+ff-root of unity in Fqepff.
                                        ff
     o The inclusion p1+2fl+  GL pfl(qep ) is as in Section 4.

  Fix now m   1 and Rm,ff,flthe image of Rff,flin GL mepff+fl(q) which sends g *
 *to the m-fold

diagonal map.

  Finally consider c = (c1, . .,.ct) a sequence of positive integers and define*
 * Fc def=(Z=p)c1o

(Z=p)c2o . .o.(Z=p)ct. Let d def=mepff+fl+c1+...+ctand denote by Rm,ff,fl,cthe *
 *image of Rm,ff,flo Fc

in GL d(q). Call a subgroup of this type as a basic subgroup of GL d(q).


Theorem 5.2 ([1]). Fix V an n-dimensional Fq-vector space. Let G = GL (V ), and*
 * R be a

p-radical subgroup of G. Then there exist decompositions

                              W = V0   V1   . . .Vs,

                              R = R0 x R1 x . .x.Rs,


such that R0 is the trivial subgroup of GL (V0), and Riare basic subgroups of G*
 *L (Vi) for i   1.


  Now we are able to give the Sylow p-subgroup in terms of e, (the order of q m*
 *odulo p), l

(the p-adic valuation of qe - 1) and the coefficients a0, . .,.ak, all of them *
 *0   ai   (p - 1)

such that:

                            [n=e] = a0 + a1p + . .+.akpk.


Lemma 5.3. Sn,q, a Sylow p-subgroup of GL n(q), is given by the following const*
 *ruction:

                def    l       (k)      a       l       (k-1)     a
            Sn,q=  (Z=p o Z=po . .o.Z=p) kx (Z=p o Z=po  . .o.Z=p) k-1x . . .

                   . .x.(Z=plo Z=p)a1x (Z=pl)a0.

where e, l and a1, . .,.ak are defined above.

  Moreover, if (Sn,q, Fn,q) is the saturated fusion system induced by GL n(q) o*
 *ver Sn,q, then

Out Fn,q(Sn,q) is isomorphic to
                      k                           k-1
      (Z=e x Z=(p - 1) ) o  ak x  (Z=e x Z=(p - 1)   ) o  ak-1 x . .x. (Z=e) o *
 * a0 .


Proof.This combination of direct products and wreath products is included in GL*
 * e[n=e](q)

GL n(q). Checking the orders of Sn,qand the Sylow p-subgroup of GL n(q) we have*
 * finished.

  Finally the outer automorphisms group is computed in [1, Section 4].


  For fixed p and q we can compute e, and the Sylow p-subgroup and the fusion d*
 *oes not

change between GL n(q) and GL e[n=e](q). Sometimes we will consider that we are*
 * working in

rank multiples of e and we will write in GL em(q) instead of GL n(q).


Notation 5.4. Consider Sn,qa Sylow p-subgroup of GL n(q) as computed in Lemma 5*
 *.3, and

(Sn,q, Fn,q, Ln,q) the corresponding p-local finite group.


  Let us begin now with the first non-trivial case:


Lemma 5.5. Let p be an odd prime and q a prime power such that p|(q - 1).  Cons*
 *ider

(Sp,q, Fp,q) as in Notation 5.4. Then Out 0Fp,q(S) = OutFp,q(S).


            EXOTIC NORMAL FUSION SUBSYSTEMS OF GENERAL LINEAR GROUPS          11



Proof. To simplify the notation, consider in this proof (S, F) def=(Sp,q, Fp,q)*
 *. In this case, by

Lemma 5.3 we have Out F(S) ~=Z=(p - 1). We can see that oe defined in Equation *
 *(2)is a

generator, so we must check that oe 2 Out0F(S).

  Consider R0,1as in Equation (6). We get that R0,1is isomorphic to a central e*
 *xtension of

p1+2+, generated by A and B as defined in Equations (3)and (4), by Z=pl, the p-*
 *primary part

in the center of GL p(q).

  A direct computation tells us that R0,1is F-centric.

  Equation (2)gives us the action of A and B:

                                       -1         -1    i
                           oeAoe-1 = Ai  and oeBoe   = B  ,


where i is a (p - 1)-root in Fp. So oe restricts to an automorphism of R0,1whic*
 *h, as element
                                                    -10
in OutF (R0,1), must be considered as the matrix   i0 i .

  Use now that in this case, [1, Section 4] tells us that Out F(R0,1) ~=SL2(p),*
 * and now apply

the fact that SL2(p) is generated by its elements of order p.


Proposition   5.6.  Consider  (Sn,q, Fn,q)  as  in  Notation  5.4,   with  n   *
 *    ep.

There is a subgroup H in OutFn,q(Sn,q) such that Out Fn,q(Sn,q)=H ~=Z=e and Out*
 * 0Fn,q(S)   H.


Proof. Let us consider the form of OutFn,q(Sn,q) given in Lemma 5.3.

  Observe that all the elements in Out Fn,q(Sn,q) restrict to automorphisms of *
 *the maximal

torus T[n=e]pl, and that different elements in Out Fn,q(Sn,q) give different re*
 *strictions.  This

implies that if F0 is the minimal saturated fusion subsystem of (Sn,q, Fn,q) of*
 * index prime to

p over S, then


              Out Fn,q(Sn,q)= Out0Fn,q(Sn,q)~=OutFn,q(Sn,q)= OutF0(S) ~=

                                        ~= OutFn,q(T[n=e])= Out  (T[n=e]) .    *
 * (7)
                                                    pl         F0  pl


We have that Out Fn,q(T[n=e]pl) ~=Z=e o  [n=e]. Now consider Op0(Out Fn,q(T[n=e*
 *]pl)), the subgroup

generated by all the elements of order a power of p. As p   [n=e], there are el*
 *ements of order p

in  [n=e], and as Op0(Out Fn,q(T[n=e]pl)) is a normal subgroup, it musts contai*
 *n all the alternating

group A[n=e]   [n=e]and its normal closure in (Z=e)[n=e]o A[n=e], whichPis (Z=e*
 *)[n=e]-1o A[n=e]

((Z=e)[n=e]-1  (Z=e)[n=e]is the kernel of the map (a1, . .,.a[n=e]) 7!   ai).

  To finish the proof, take the restriction of oe, which by Lemma 5.5 is in Out*
 * 0Fn,q(S), and

which gives an odd permutation of  [n=e], getting that


                       HT def=(Z=e)[n=e]-1o  [n=e]  OutF0(T[n=e]pl) .


But, by Equation (7), Out 0Fn,q(S) must be contained in H defined as the kernel*
 * of the mor-

phism:

                        OutFn,q(S) ! OutFn,q(T[n=e]pl)=HT ~=Z=e ,

and the result follows.


  From the previous proposition we get that Out Fn,q(S)= Out0Fn,q(S) has at mos*
 *t e ele-

ments.  The rest of the section is dedicated to prove that there is an element *
 *of order e

in OutFn,q(S)= Out0Fn,q(S).


12                                  ALBERT RUIZ


  As we are interested just in the Fn,q-centric, Fn,q-radical subgroups, we can*
 * remove some

subgroups in the list:


Lemma 5.7. If m > 1 then Rm,ff,fl,cis not p-centric in S.


Proof.Assume m > 1. We are considering the inclusion of Rff,flin GL mepff+fl(q)*
 * diagonally:
                                   0             1
                                      g 0  . . .0
                                   B  0 g  . . .0C

                              g 7! BB .  . .    .CC .
                                   @  .. .. ..  ..A

                                      0 0  . . .g


Fix g a non-trivial element in the center or Rff,fl. We have that
                                   0              1
                                      g  0  . . .0
                                   B  0  1  . . .0C

                               h def=BB. .  .   . CC
                                   @  .. ..  .. ..A

                                      0  0  . . .1


is an element which centralizes Rm,ff,flwhich is not in Rm,ff,fl.

  Recall now that in a semidirect product G o H, if we have an element g in the*
 * center of G

invariant under the action of H, then (g, 1) is in the centre of G o H. Apply t*
 *his argument
                                           c1+...+cr
to the element (h, h, . .,.h) 2 (Rm,ff,fl)p      , which is invariant by the ac*
 *tion of  n.


  So, from now on, we just consider the case m = 1.


Lemma 5.8. Consider ~ a pl-root of unity in Fqe. The maximal torus Tmplcan be s*
 *een as the

image of (Z=pl)m under the composition


                (a1, . .,.am ) 7! diag(~a1, . .,.~am) 2 GL m(qe)   GL em(q) .


Fix R an Fem,q-centric, Fem,q-radical subgroup in Sem,q(Notation 5.4) and consi*
 *der O the

image in GL em(q) of the subset (not a subgroup) of elements in (Z=pl)m such th*
 *at the product
                      0
a1. .a.m= 1. If _ 2 Op (Aut Fem,q(R)), then _(O \ R) = O \ R.


Proof.To simplify the notation in this proof consider (S, F) def=(Sem,q, Fem,q)*
 * Consider R an

F-centric, F-radical subgroup of GL (V ), with V an n-dimensional Fp-vector spa*
 *ce. By the

description in Theorem 5.2 we get that there is a decomposition which can be gi*
 *ven, after

reordering and grouping by isomorphism type, as:


                    V = V0   V1,1  . . .V1,m1  V2,1  . . .Vk,mk


such that R can be written as:


                           R = R0 x (R1)m1 x . .x.(Rk)mk


with Ri basic subgroups in GL (Vi,j), and such that m1d1 + . .+.mkdk = m, where*
 * di =

dim (Vi,1). For each i_2_{1, . .,.k} and j 2_{1, . .,.mi}, consider Hi,jthe ima*
 *ge of GL (Vi,j)

in GL (V ).  Consider S i,j= S \ Hi,j, and F i,jthe saturated fusion system F_S*
 *i,j(GL (Vi,j)).
             __    __       __    __
Observe that Si,j~=Si,j0and F i,j~=Fi,j0for j, j02 {1, . .,.mi}.


            EXOTIC NORMAL FUSION SUBSYSTEMS OF GENERAL LINEAR GROUPS          13


  With all this notation, we have the isomorphism:

                                     Yk

                        AutF (R)=R ~=   (Aut __Fi,1(Ri)=Ri) o  mi

                                     i=1

The action of R on itself by conjugation is included in the action of S over R,*
 * and this

preserves O, as these are permutations.
                                                                       0
  Consider now an element in Aut F(R) such that its class belongs to Op (Out F(*
 *R)).  We

have to deal with two possibilities:
                 __
     o As Ri are F i,j-centric basic subgroups, using Lemma 5.7 and [1, Section*
 * 4] all the

       elements of order a power of p in Out__Fi,1(Ri) are permutations, so the*
 *y preserve O.

     o If there exists i such that mi   p then the elements of the form ooe(o-1*
 *) are also in
         0
       Op (Out F(R)), for all oe in Ami    mi and all o 2 AutF0i,1(Ri) (Ami the*
 * alternating

       subgroup in  i). To simplify notation consider the elements in O\Si,jas *
 *(a1, . .,.adi) 2

       (Z=pl)di. But o sends (a1, . .,.adi) to (ao(1)q1, . .,.ao(di)qdi), and s*
 *o oe(o-1) permutes

       the components and multiply each one by q-1irespectively. In any case, t*
 *he product

       of the elements is


                         a1. .a.diq1. .q.diq-11. .q.-1di= a1. .a.di.


       So the product a1. .a.didoesn't change.

This implies that the elements of _(O \ R)   O, but as _ is an automorphism of *
 *R the result

follows.


  To get an element of order e in Aut 0Fn,q(Sn,q) we use the automorphism '1 de*
 *fined in

Equation (5)as follows: as we consider the case n   e, consider OE the Fq-linea*
 *r map defined

as

                                 OE def='1   Id[n=e]-1.                        *
 * (8)

Observe that OE is an element of order e.


Proposition 5.9. Consider (Sn,q, Fn,q) as in Notation 5.4, with n   e.  For eve*
 *ry Fn,q-

centric subgroup P , and j such that 1   j   e - 1, the restriction of OEj, whe*
 *re OE is defined

in Equation (8), to P is not in Hom Op0*(Fn,q)(P, Sn,q).


Proof. Consider the restriction of OEj an Fn,q-centric subgroup P .  As P is Fn*
 *,q-centric, it

contains the center of Sn,q, which can be seen as the image of the matrices ~ I*
 *din the

inclusion GL m (qe)   GL em(q). Assume now that OEj|P is in Hom Op0*(Fn,q)(P, S*
 *n,q).

  By Theorem 3.4, OEj can be written as a composition of restriction of automor*
 *phisms OEi2
  0
Op (Aut Fn,q(Ri)) (i 2 {1, . .,.k}), for Ri Fn,q-centric, Fn,q-radical subgroup*
 *s. Consider now

ae Id, ae a primitive p-root of unity in Fqe, which is an element in O \ P (O a*
 *s defined in Lemma

5.8). This implies OEj(ae Id) = OEk O OEk-1O . .O.OE1(ae Id) 2 O.
                                                   -1                          *
 *   j
  A direct computation show us that OE(ae Id) = aeq  Id1 ae Id[n=e]-1, and then*
 * OE (ae Id) =
   -j                                                   -j
aeq  Id1 ae Id[n=e]-1. So this element is not in O, as q   6= 1 mod p.

  So we get a contradiction which comes from assuming that


                             OEj|P 2 Hom Op0*(Fn,q)(P, Sn,q) .


14                                  ALBERT RUIZ





Theorem 5.10. Let p be an odd prime and q prime power such that p - q. Consider*
 * e the

multiplicative order of q modulo p. Fix Sn,qa Sylow p-subgroup of GL n(q) and (*
 *Sn,q, Fn,q) the

induced saturated fusion system. Then,
                                           (
                                             (Z=e) o  [n=e]if e   n < ep,
                OutF (Sn,q)= Out0F(Sn,q) ~=
                                             Z=e          if ep   n.


Proof.We have to consider n   e to have a nontrivial Sylow p-subgroup in GL n(q*
 *).

  For n < ep we have that Sn,q~=(Z=pl)[n=e], which is abelian. So the only F-ce*
 *ntric subgroup

is the total and Out0Fn,q(Sn,q) is trivial.

  When n   ep, use Proposition 5.6 to see that the quotient is at most Z=e and *
 *Proposition

5.9 implies the result.


           6. Exotic fusion subsystems in the general linear group


  Fix as before p a prime, q a prime power prime to p. Fix e the multiplicative*
 * order of q

modulo p. Consider (Sn,q, Fn,q, Ln,q) the saturated fusion system of GL n(q) at*
 * the prime p,

with n   e.


Proposition 6.1. (a) For each r dividing e there is a p-local finite group (Sn,*
 *q, Fn,q,r, Ln,q,r),

    such that Fn,q,ris of index prime to p in Fn,qover Sn,qand Out Fn,q,r(Sn,q)*
 * is a subgroup

    of index r in OutFn,q(Sn,q).

(b) If n   ep, there is just one (Sn,q, Fn,q,r, Ln,q,r) satisfying (a).

(c) Up to homotopy equivalence, there is a fibration


                            |Ln,q,r| -! |Ln,q| -! B(Z=r) .


Proof.For e   n < ep consider the saturated fusion subsystem of index prime to *
 *p corre-

sponding to the kernel of the group epimorphism from


                        Out F(Sn,q)= Out0F(Sn,q) ~=(Z=e) o  [n=e]


to Z=r defined as

                      (Z=e)[n=e]o  [n=e]-!  Z=r
                                            e
                      (b1, . .,.b[n=e];7oe)!_(b1 + . .+.b[n=e]) .
                                            r
getting (a) and (c) for this case, applying [2, Theorem 5.5].

  For n   ep we have Out F(Sn,q)= Out0F(Sn,q) ~=(Z=e), so (a), (b) and (c) foll*
 *ows directly

again from [2, Theorem 5.5].


  We proceed now to identify these saturated fusion subsystems with the ones st*
 *udied in [5,

Section 11].

  Let p be an odd prime and r   1, e   1 natural numbers such that r|e|(p - 1).*
 * Consider

G(e, r, m) the subgroup of GL m(Z^p) as the subgroup generated by:


              A(e, r, m) = {diag(a1, . .,.am ) | aei= 1 and (a1. .a.m)e=r= 1}


and the matrices corresponding to permutations in the coordinates. As a group, *
 *we have that

G(e, r, m) = A(e, r, m) o  n.


            EXOTIC NORMAL FUSION SUBSYSTEMS OF GENERAL LINEAR GROUPS          15


  Consider now m   1 and let X(e, r, m) be the p-compact group which realizes t*
 *he pseu-

doreflection group G(e, r, m)   GL m(Z^p) (see [8] and [9] for further informat*
 *ion). If m   2

these are called the Generalized p-adic Grassmannians and we are interested in *
 *their finite

Chevalley version. That is the space BX(e, r, m)(q) defined as the pullback dia*
 *gram:


                     BX(e, r, m)(q)__________//BX(e, r, m)

                           |                       |
                           |                       |
                           fflffl| 1x'q            fflffl|
                      BX(e, r, m)______//BX(e, r, m) x BX(e, r, m)


where   is the diagonal map and 'q is the unstable Adams operation of exponent *
 *q, where

q is a p-adic unit.

  So if we consider q a power of a prime not divisible by p, it is a p-adic uni*
 *t and the previous

construction makes sense. Consider e the multiplicative order of q modulo p. In*
 * this case we

have the following equivalences up to p-completion [5, Remark 11.1]:

     o BX(1, 1, n)(q) ' BGL n(q).

     o BX(1, 1, n)(q) ' BX(1, 1, e[n=e])(q).

     o If q0 is a another prime power such that p|(q0- 1) and  p(q0- 1) =  p(qe*
 * - 1) then

       BX(1, 1, n)(q) ' BX(e, 1, [n=e])(q0). Observe that we can take q0= qe.

Where [n=e] is the greater integer less or equal than n=e.

  Before the main statement of the section we need to compute some centralizers.


Lemma 6.2. Consider (Sn,q, Fn,q,r, Ln,q,r) as in Proposition 6.1, with n   e, a*
 *nd V a non-

trivial elementary abelian p-subgroup contained in Sn,q. Then

(a) V is Fn,q,r-conjugate to a subgroup of the maximal torus T[n=e]pl.

(b) For V contained in T[n=e]pl, consider the centralizer p-local finite group


                            (CSn,q(V ), CFn,q,r(V ), CLn,q,r(V ))


    defined in [3, Proposition 2.5]. Then there are integers m0, m1, . . . , ms*
 * such that


                 |CLn,q,r(V )| ' |Lem0,q,r| x BGL m1(qe) x . .x.BGL ms(qe)


    up to p-completion. Moreover em0 < n and [n=e] = m0 + m1 + . .+.ms.


Proof. Consider V  a nontrivial elementary abelian p-group contained in S.  Con*
 *sider the

saturated fusion system (Sn,q, FSn,q(GL [n=e](qe))) which is contained in (Sn,q*
 *, Fn,q,r) for all

r|e.  Then we can use the fact that V  is FSn,q(GL [n=e](qe))-conjugate to a to*
 *ral subgroup,

getting (a).

  To compute the centralizer fusion system, consider the point-wise stabilizer *
 *of V in


                            AutFn,q,r(T[n=e]pl) = G(e, r, [n=e]),


which is isomorphic to G(e, r, m0) x  m1 x . .x. ms. Note that as V is a nontri*
 *vial group,

m0 < [n=e].

  We can see that CFn,q,r(V ) is a saturated fusion system over CSn,q(V ), whic*
 *h can be written

as CSn,q(V ) ~=Sem0,qxSm1,qex. .x.Sms,qe, where we are using that Lemma 5.3 ide*
 *ntifies Sm,qe

and Sem,q. Also CFn,q,r(V ) contains Fem0,q,rx FSm1,qe(GL m1(qe)) x . .x.FSms,q*
 *e(GL ms(qe)),


16                                  ALBERT RUIZ


as all the morphisms in this last saturated fusion system centralize V and are *
 *contained in

Fn,q,r.

  Considering now that Fn,q,ris a fusion subsystem of Fn,q, the morphisms in th*
 *e centralizer

of V in Fn,q,rare again morphisms in the centralizer of V in Fn,q.

  The centralizer p-local finite group of V in Fn,qis computed in [5, Propositi*
 *on 11.2], and

its classifying space is mod p equivalent to BGL em0(q) x BGL m1(qe) x . .x.BGL*
 * ms(qe), and

also has the same Sylow p-subgroup as CFn,q,r(V ).

  With all this data we get that CFn,q,r(V ) is a saturated fusion subsystem of*
 * index prime to

p in CFn,q(V ) such that in the maximal torus Aut CFn,q,r(V()T[n=e]pl) is an in*
 *dex r subgroup of

Aut CFn,q(V()T[n=e]pl), and so


           CFn,q,r(V ) ' Fem0,q,rx FSm1,qe(GL m1(qe)) x . .x.FSms,qe(GL ms(qe))*
 * .


So (b) follows considering the centric linking systems associated to these satu*
 *rated fusion

systems.


Theorem 6.3. For each r dividing e and n   e consider (Sn, Fn,q,r, Ln,q,r), the*
 * p-local finite

group from Proposition 6.1. Then


                             |Ln,q,r| ' BX(e, r, [n=e])(qe)


up to p-completion.


Proof.We proceed by induction on n, beginning by the smallest considered case, *
 *that is

n = e.

  For GL e(q) we have that the p-local finite group (Se,q, Fe,q,r, Le,q,r) is c*
 *haracterized by

Se,q= Z=pl, and Aut Fe,q,r(Se,q) = Z=h, where h def=e=r.  So it corresponds to *
 *the fusion

system of Z=plo Z=h, where h def=e=r.  The Sylow p-subgroup of BX(e, h, 1)(q) i*
 *s also

isomorphic to Z=pl, with outer automorphism group G(e, h, 1) = Z=h, so both are*
 * the same

p-local finite group.

  Now assume that the result is true for any n0  n.

  To proceed with the induction argument we use the centralizer decomposition f*
 *rom [3,

Theorem 2.6].

  Fix V    S an elementary abelian fully Fn,q,r-centralized subgroup.  If neces*
 *sary we can

conjugate it to get V contained in the torus, and still being fully Fn,q,r-cent*
 *ralized.

  Consider (CSn,q(V ), CFn,q,r(V ), CLn,q,r(V )) the centralizer p-local finite*
 * group, whose classi-

fying space is, by Lemma 6.2, homotopy equivalent up to p-completion to


                      |Lem0,q,r| x BGL m1(qe) x . .x.BGL ms(qe) ,


with em0 < n. By induction hypothesis we have, up to p-completion, |Lem0,q,r| '*
 * BX(e, r, m0)(qe)

and so
           |Lem0,q,r| x BGL m1(qe) x . .x.BGL ms(qe) '
                                                                               *
 * (9)
                          BX(e, r, m0)(qe) x BGL m1(qe) x . .x.BGL ms(qe) .

Now consider the centralizer decomposition from [3, Theorem 2.6], obtaining


                            hocolim-------!|CLn,q,r(V )| ' |Ln,q,r|
                             V 2(Fe)op


            EXOTIC NORMAL FUSION SUBSYSTEMS OF GENERAL LINEAR GROUPS          17


while using Equation (9)and [5, Proposition 11.3] we also get


                        hocolim-------!|CLn,q,r(V )| ' BX(e, r, [n=e])(qe)
                        V 2(Fe)op

getting the result.


  We finally get one of the results which motivated this work.


Remark 6.4. Consider p a prime, q a prime power such that e, the order of q mod*
 *ulo p, is

bigger than 2. Consider r a divisor of e such that r > 2. Then, by Proposition *
 *6.1 we have

the fibration (up to homotopy equivalence):


                            |Ln,q,r| -! |Ln,q| -! B(Z=r) .


By Theorem 6.3, we have

                             |Ln,q,r| ' BX(e, r, [n=e])(qe)

up to p-completion, and by [5, Proposition 11.5], when r > 2 and n   ep, these *
 *are classifying

spaces of exotic p-local finite groups. So we get examples of extensions of p-l*
 *ocal finite groups

where two of them correspond to finite groups and the third one is an exotic p-*
 *local finite

group.


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  Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, 08193 Cerda*
 *nyola
del Vall`es, Spain.
  E-mail address: Albert.Ruiz@uab.cat