On the Dirichlet polynomial of finite group of Lie type

On the Dirichlet Polynomial of Finite Groups of Lie Type.

ERIKADAMIAN(*) - ANDREALUCCHINI(*)

Dedicated to Guido Zappa on his 90th birthday

ABSTRACT- For a given finite groupGthere exists a uniquely determined Dirichlet polynomialPG(s) with the property that fort2Nthe numberPG(t) coincides with the probability of generatingGbytrandomly chosen elements. We discuss whether the isomorphism type of a simple groupGcan be determined by the knowledge ofPG(s):

1. Introduction.

For a given finite group G one may define a sequence of integers fa_n(G)g_n2N as follows:

a_n(G)ˆ X

jG:Hjˆn

m_G(H):

Herem_Gdenotes the MoÈbius function defined on the subgroup lattice ofG.

In particular, one hasm_G(G)ˆ1 andm_G(H)ˆ P

H<Km_G(K) for anyH<G.

Let

P_G(s)ˆX

n2N

a_n(G) n^s

be the Dirichlet generating function associated with the sequence fan(G)g_n2N. The Dirichlet polynomial PG(s) gives a great amount of in- formation aboutG:In [13] P.Hall observed that for anyt2N the series P_G(t) gives the probability thattrandomly chosen elements ofGgenerate G. Moreover (see for example [3]), the complex function P_G(s), and in (*) Indirizzo degli AA.: UniversitaÁ di Brescia, Dipartimento di Matematica, Via Valotti, 25133 Brescia, Italy.

E-mail: [email protected] E-mail: [email protected]

particular its value at -1, can be used to investigate the topological prop- erties of the poset of proper cosets ofG:

A problem that has been tackled from various points of view ([10], [9], [6]) concerns the study of which properties ofP_G(s), as an element of the ring of Dirichlet polynomials with integer coefficients, reflect on properties of the groupG. Recently [8] we proved that the knowledge ofP_G(s) allows us to decide whether the factor groupG=Gis a simple group or not: ifG1is simple andP_G2(s)ˆP_G1(s);thenG2=(G2) is simple. We conjecture that a stronger result is true: if G₁ is simple and P_G2(s)ˆP_G1(s) then G₂=G₂G₁. This has been proved for alternating simple groups [7], so the aim of this paper is to investigate the case of simple groups of Lie type. The main result we will obtain is that if we know thatGis a simple group of Lie type over a field of characteristicp;then with the help of the functionP_G(s) we can determine the order of a Sylowp-subgroup ofG. This result will be employed to prove that a sporadic simple group can be identified from its Dirichlet polynomial, and also to show that ifG₁andG₂are non isomorphic simple groups of Lie type defined in the same characteristic, then P_G2(s)6ˆP_G1(s):

2. Known results and open questions.

LetGbe a finite simple group. We want to discuss about the properties ofGthat can be deduced from the knowledge of the Dirichlet polynomial P_G(s):The simplest thing to do is to consider the set

y(G)ˆ fn2Nja_n(G)6ˆ0g:

Ifn2y(G) thenGmust have a subgroup of indexn;moreover the smallest n2y(G) with n>1 coincides withm(G); the smallest index of a proper subgroup ofG:The main obstacle when we work withy(G) is that even when we know that G has a subgroup H with index n we are not sure than n2y(G);in fact, in order to decide whethera_n(G)6ˆ0 we should focus on the set of subgroupsHGwith indexnand such thatm_G(H)6ˆ0;but here we meet another problem as the suma_n(G)ˆ P

jG:Hjˆnm_G(H) can be zero even if the terms are different from zero. A case in which it is easy to deduce that a_n(G)6ˆ0 is when we know thatGhas a maximal subgroup of indexnand there is no maximal subgroup with index a proper divisor ofn(in that case a_n(G) is precisely the number of maximal subgroups of indexn). One can expect that the sety(G) is large enough to give good hints concerning the

order of the group G:Let us formulate two conjectures in this direction.

Define the probabilistic order ofGas follows:

po(G)ˆl:c:m:fnjn2y(G)g

Clearly po(G) dividesjGj;our first conjecture is that po(G)ˆ jGj;a weaker conjecture is that we can deduce fromy(G) which are the prime divisors of jGj, namely:p(po(G))ˆp(G): Both these conjectures are open questions and it seems a difficult but intriguing task to prove something in this di- rection without a heavy use of the classification of finite simple groups and of their maximal subgroups.

In [9] and [7] it is proved that an analysis of y(G) allows to decide whetherGis an alternating group; more precisely we have:

THEOREM1. Let G be a finite nonabelian simple group. Set nˆm(G) and let p be the minimal prime number which divides n;

(1) if nˆp then: GAlt (n)if and only if(n 2)!2y(G);

(2) if n>p and n2 f6,24g= then: GAlt (n) if and only if a_n(G)ˆ n and n

p is the smallest integer iny(G) which is different from 1 and not divisible by n;

(3) if nˆ6then: GAlt (6)if and only if a₆(G)ˆ 12;

(4) if nˆ24then: GAlt (24)if and only if253, 7592=y(G):

Hence for the rest of the paper we shall restrict our attention to sporadic simple groups and simple groups of Lie type.

3. Dealing with groups of Lie type.

The problem of recognizing a simple group Gfrom its Dirichlet poly- nomialPG(s) is still open. In this section we shall prove a result which turns out to be useful for the analysis of this problem for groups of Lie type.

Indeed we shall show that if we know thatGis a simple group of Lie type over a field of characteristicp; then we may use the polynomialP_G(s) in order to determine the order of a Sylowp-subgroup ofG:

We need to define some other Dirichlet polynomials associated withG and its subgroups.

LetRbe the ring of Dirichlet polynomials with integer coefficients; for any finite set of prime numberspwe may define a ring endomorphism ofR as follows:

h_p: R ! R f(s)ˆX¹

nˆ1

an

n^s 7! f^(p)(s)ˆX¹

nˆ1

bn

n^s where

bnˆ an if nis ap⁰-number 0 otherwise:

(

We are mainly interested inP^(q)_G(s), withqa prime number.

Moreover, for any subgroup K of G, we may define a Dirichlet poly- nomial as follows:

P_G(K;s)ˆX

n2N

an(G;K)

n^s where an(G;K)ˆ X

jG:Hjˆn;

KHG

m_G(H):

LEMMA2. Let P be a Sylow p-subgroup of a finite group G, p a prime number; suppose that each maximal subgroup of G which contains P contains also N_G(P):Then

P^(p)_G (s)ˆP_G(P;s 1)ˆP_G(N_G(P);s 1):

PROOF. First we claim that m_G(H)jNH(P)j ˆm_G(H)jN_G(P)j for each subgroup PHG: Indeed, either m_G(H)ˆ0 or H can be written as intersection of maximal subgroups, say M₁;. . .;M_t (see [14]); as PHM_i; by hypothesis we get that N_G(P)M_i, which implies N_G(P)M₁\ \M_tˆHand N_G(P)ˆN_H(P): Now set V_pˆ

ˆ fHGj jHj_pˆ jGj_pg, then P^(p)_G (s)ˆ X

H2Vp

m_G(H) jG:Hj^s ˆ

ˆ X

Q2Syl_p(G)

X

QH

m_G(H)

jG:Hj^s 1 jH:N_H(Q)jˆ

ˆ 1 jNG(P)j

X

PH

m_G(H)jN_H(P)j jG:Hj^{s 1} ˆ

ˆ 1 jN_G(P)j

X

PH

m_G(H)jN_G(P)j jG:Hj^{s 1} ˆ

ˆX

PH

m_G(H)

jG:Hj^{s 1}ˆP_G(P;s 1):

This proves the first equality in our statement. The other one, P_G(P;s 1)ˆP_G(N_G(P);s 1); is again an immediate consequence of the previous remark that if m_G(H)6ˆ0 andPH, then N_G(P)H:

p THEOREM3. Suppose that G is a finite group of Lie type defined over a field of characteristic p and let U2Syl_p(G):ThenjP^(p)_G(0)j ˆ jUj.

PROOF. A finite groupGof Lie type over the fieldF_q,qˆp^f;can be constructed starting from a connected reductive algebraic groupXdefined over an algebraically closed field of characteristic p and considering the subgroupGˆX^Fof fixed points under a Frobenius mapF:LetBbe anF- stable Borel subgroup ofX:The unipotent radicalU ofB^F is a Sylowp- subgroup ofG^FandN_G(U)ˆB^F. As it is well known, a maximal subgroup ofG^Fwhich containsUshould containB^F, hence it is a maximal parabolic subgroup ofG^F, so we can apply the previous lemma in order to deduce that

P^(p)_G(0)ˆPG(B^F; 1)ˆ X

B^FH

m_G(H)jG:Hj:

To the mapFa symmetryron the Dynkin diagram ofXis associated (ris trivial in the untwisted case). LetI:ˆ fO1;. . .;O_kgbe the set of ther-or- bits on the nodes of the Dynkin diagram. IfJI, thenJˆ[

j2J

O_jis ar- stable subset of the set of nodes of the Dynkin diagram and one may as- sociate anF-stable parabolic subgroupPJofXwithJ. Moreover, the map J7!P^F_Jis an isomorphism between the latticeP(I) of subsets ofIordered by inclusion, and the lattice of subgroups ofGcontainingB^F. In particular, m_G(P_J)ˆm_P(I)(J)ˆ( 1)^{k jJj}(see [20], 3.8.3). As described in [4], Ch. 9, to any subset Jof I, a parabolic subgroup W_J of the Weyl groupW^F and a polynomialPWJ(x) are associated with the property thatPWJ(q)ˆ jP^F_Jj:So one has

P^(p)_G(0)ˆ X

B^FH

m_G(H)jG:Hj ˆX

JI

m_G(P^F_J)jG:P^F_Jj

ˆX

JI

( 1)^{k jJj}jG:P^F_Jj ˆ( 1)^kX

JI

( 1)^jJj PW(q) P_WJ(q)

By a theorem of Solomon (see 9.4.5. and Ch. 14 in [4]) X

JI

( 1)^jJj P_W(q) PWJ(q)

ˆ jUj

so we conclude thatP^(p)_G(0)ˆ( 1)^kjUj: p

COROLLARY4. LetSbe a simple group of Lie type defined over a field K; if we know that the characteristic ofK isp;then we may determine from the Dirichlet polynomialPS(s)the order of a Sylowp-subgroup ofS:

PROOF. Let S ˆ f²F₄(2)⁰;G₂(2)⁰;²G₂(3)⁰;B₂(2)⁰g: If S2 S= then there exists a finite group of Lie typeGwithSˆG=Z(G); moreoverpdoes not dividejZ(G)jand, by Theorem 3,P^(p)_G(0)ˆP^(p)_S (0) is the order of a Sylow p-subgroup of S: On the other hand a direct computation shows that jP^(p)_S (0)j ˆpjUjwhenS2 SandU is a Sylowp-subgroup ofS:Note that m(B₂(2)⁰)ˆ6; m(G₂(2)⁰)ˆ28; m(²G₂(3)⁰)ˆ9;m(²F₄(2)⁰)ˆ1600: Now let Sbe a simple group of Lie type defined over a field of characteristic p:

From the series PS(s) we recover the value of m(G): If (p;m(S))2=

=

2f(2, 6);(2, 28);(2, 1600); (3, 9)g, thenS62 SandjP^(p)_S (0)jcoincides with the order of the Sylow p-subgroup of S. Ifpˆ2 andm(S)ˆ28 then either SˆB₃(2) and jP⁽²⁾_G(0)j ˆ2⁹ or SˆG₂(2)⁰ and jP⁽²⁾_G(0)j ˆ2⁶: If pˆ2 and m(S)ˆ1600 then Gˆ²F₄(2)⁰; if pˆ2 and m(S)ˆ6 then GˆB₂(2)⁰; if

pˆ3 andm(S)ˆ9 thenGˆ²G2(3)⁰: p

4. Connection between Theorem 3and the Solomon-Tits Theorem.

In the previous section we proved Theorem 3 by computing directly P_G(B^F; 1), which is possible as we can compute the index and the MoÈbius function of any parabolic subgroup ofG. We would like now to present a different proof for the same result; this is less immediate and direct, but it makes evident the connection between the study of the Dirichlet polynomial PG(s) and some topological properties of the poset of proper cosets inG.

We first revise and generalize a well-known result of K. S. Brown [3].

LetKbe a proper subgroup of a finite groupG:We define two posets:

the first one, C ˆ C(G;K); consists of the proper cosets Hx (KH<

<G;x2G) ordered by inclusion. The secondCˆ C(G;K) is the subposet ofCconsisting of the cosetsHxwhereHsatisfies the additional property of being intersection of maximal subgroups ofG:As it is well known, we can apply topological concepts to a posetPby using the simplicial complexD(P) (the order complex ofP) whose simplices are the finite chains inP:

LEMMA5. The complexesD(C)andD(C)are homotopy equivalent.

PROOF. We recall the following criterion due to Quillen ([18] Proposi- tion 1.6): if there exists an order preserving mapf :X!Y between two

posets, with the property that for any y2Y the lower fiber f=yˆ

ˆ fx2Xjf(x)ygis contractible, then the complexesD(X) andD(Y) are equivalent. In our case we may define an order preserving map f :C ! C by sending Ux to Ux; being U the intersection of the maximal sub- groups of G containing U: For any Vx2 C the lower fiber f=Vxˆ

ˆ fUyjy2VxandKUVg is contractible as Vx is a least upper

bound for f=Vx: p

IfG is ann-dimensional complex, the Euler-PoincareÁ characteristic of G is the integer x(G)ˆ P

0qn( 1)^qaq;where aq is the number ofq-sim- plices of G: The reduced Euler-PoincareÁ characteristic is defined as:

~

x(G)ˆx(G) 1:

Generalizing a result due to Brown in the particular case Kˆ1; we prove that the Euler-PoincareÁ characteristic of the complexes D(C) and D(C) can be computed with the help of the functionPG(K;s):Indeed one has:

PROPOSITION6. PG(K; 1)ˆ ~x(D(C(G; K)))ˆ ~x(D(C(G; K))):

PROOF. SinceD(C(G;K)) andD(C(G;K)) are equivalent, it suffices to prove the first equality. LetCbe the poset obtained by adding toC(G;K) a greatest element (which we may well take asG) and a least element (that we denote by 0). Recall that the MoÈbius functionm_Passociated to a posetPhas the property that if a<b then m_P(a;b) is the number of chains of even length in the interval (a;b)ˆ fc2 P ja<c<bg minus the number of chains of odd length (here we include the empty chain, which we agree to consider of length -1); this implies that m_P(a;b)ˆ~x(D(a;b)):In our parti- cular case we obtain mC(0;G)ˆ~x(D(C(G;K)): Now let Hx2 C(G;K); the interval [H;G] in the subgroup lattice of Gis isomorphic to the interval [Hx;G] inCvia the mapU!Ux;thusmC(Hx;G)ˆm_G(H) and

P_G(K; 1)ˆ X

KHG

m_G(H)jG:Hj ˆm_G(G)‡ X

Hx2C(G;K)

mC(Hx;G)ˆ

ˆmC(G;G)‡ X

Hx2C(G;K)

mC(Hx;G)ˆ mC(0;G)ˆ ~x(D(C(G;K)):

This concludes our proof. p

We shall assume for the remaining part of this section thatGis a finite group of Lie type of rankn; as it is well knownGadmits a (B;N)-pair with Ba Borel subgroup ofG:LetfM₁;. . .;M_ngbe the set of maximal parabolic

subgroups of G containing B: The Tits building T(G;B;N) of G is the simplicial complex whose vertices are the cosets Mix with x2G and 1in and whose simplices are collections of vertices with nonempty intersection. If p is the characteristic of the underlying field of G, then UˆO_p(B) is a Sylowp-subgroup ofG;and the following holds.

PROPOSITION7. The complexes D(C(G; U))andT(G;B; N)are homo- topy equivalent.

PROOF. Recall that ifBis a complex, then one can consider the faced posetP(B), consisting of the simplices ofBordered by inclusion; the order complex D(P(B)) (the first barycentric subdivision of B) is homotopy equivalent toB:Moreover ifPis a poset, thenD(P)ˆD(P^op), beingP^opthe dual poset ofP:These remarks together with Lemma 5, implies that our statement is proved if we can show thatD(P(T(G;B;N))) andD(C(G;U)^op) are equivalent. By definition an element of P(T(G;B;N)) is a set fM_i1x;. . .;M_irxg with 1i₁< <i_rn; on the other hand the ele- ments ofC(G;U) are cosetsHxwhereHis intersection of maximal sub- groups containingU; since such anHcan be written in a unique way as intersection of maximal parabolic subgroups in fM1;. . .;Mng we obtain that the mapfM_i1x;. . .;M_irxg 7!(M_i1\ \M_ir)xin an order preserving bijection between the posetsP(T(G;B;N)) andC(G;U)^op. p

COROLLARY8. P_G(U; 1)ˆ ~x(T(G;B; N)):

So we may computeP_G(U; 1) with the help of the celebrated Borel- Tits Theorem [19], which asserts that the Tits buildingT(G;B;N) has the homotopy type of a wedge ofjUjspheres, each one of dimension (n 1).

Since the reduced Euler-PoincareÁ characteristic of a wedge oft r-dimen- sional spheres is ( 1)^rtwe get:

COROLLARY9. P_G^(p)(0)ˆP_G(U; 1)ˆ( 1)ⁿjUj.

5. Consequences of Theorem 3.

PROPOSITION10. IfGis a finite simple group of Lie type over a field of characteristicp;thenp2p(po(G)):

PROOF. By Theorem 3, we have thatP^(p)_G(0)6ˆ0;while, by the definition of the MoÈbius function, P_G(0)ˆ P

HGm_G(H)ˆ0: This implies P_G(s)6ˆ

6ˆP^(p)_G (s);which is possible only when there exists a positive integerndi-

visible bypwithan(G)6ˆ0: p

It is useful to define the following set:

~y(G) is the set of n2N with the following property: G contains a maximal subgroup M of index n but it does not contain any proper subgroupHsuch thatjG:Hjis a proper divisor ofn:

In this section we will make a large use of the following fact (already recalled in section 2):~y(G)y(G):

THEOREM11. Let G be a sporadic simple group; if H is a finite simple group with P_G(s)ˆP_H(s), then GH:

PROOF. By Theorem 1,Hcannot be an alternating group. Suppose that HˆLn(q) is a group of Lie type of ranknover a fieldFqof characteristicp;

we know thatm(G)ˆm(H);and the possible values form(G) andm(H) are listed in Table 1; for any familyLof groups of Lie type, this table provides a function f(L;n;q) such that m(L_n(q))ˆf(L;n;q): So we have to check whether there are some choices ofL;n;qfor whichf(L;n;q)ˆm(G):The key tool here is that thep-adic expansion off(L;n;q) is of a very particular and recognizable shape (for example the first and last digits are equal to 1 and the second is 0 or 1); moreover by Proposition 10,p2p(po(H))ˆp(po(G));

hencepmust be a prime divisor ofjGj;so what we have to do is to write thep- adic expansion ofm(G) for any prime divisorpofjGjand check whether for some choice ofL;n;q, withqap-power, thep-adic expansions ofm(G) and of f(L;n;q) are the same. This is almost never the case; in fact there are only two possibilities: (G;H)2 f(M11;A1(11));(M24;A1(23))g. In the first case one can check thatm_M11(1)6ˆ0;sojM₁₁j ˆ79202y(M₁₁);while 79202=y(A₁(11)) (as 7920 does not divide jA₁(11)j ˆ660): this is enough to conclude that PM11(s)6ˆPA1(11)(s):In the second case, one can notice that 17712~y(M24) and does not dividejA1(23)j ˆ6072:To conclude our proof it remains to consider the case whenGandHare both sporadic simple groups. From Table 1, we have that if G and H are non isomorphic sporadic simple groups with m(G)ˆm(H), thenfG;Hg ˆ fJ₂;HSg:But we can conclude thatP_J2(s)6ˆ

6ˆP_HS(s) as 1762~y(HS) and 176 does not dividejJ₂j: p For the remaining part of this paper our attention will be restricted to finite simple groups of Lie type. It is well known, see [1], [16], that ifGand Hare non isomorphic simple groups of the same order, thenGandHei-

TABLE1. Minimal index for sporadic and Lie groups.

G m(G) G m(G)

M11 11 M12 12

M22 22 M23 23

M24 32³ J4 11²29313743

J2 2²5² J3 2²3⁴19

J1 2719 Co1 2³3³5713

Co2 2²5²23 Co3 2²323

Fi22 23³513 Fi23 3⁴1723

Fi⁰₂₄ 2³3³7²29 Ly 2²3⁴113767

McL 5²11 He 2²37³

Ru 2²5729 O'N 2³3²51131

Suz 23⁴11 B 2³3⁴5⁴233147

HS 2²5² Th 2³3⁵5³1931

HN 2⁵35⁴19 M 3⁷14⁴5565²29415971

G2(3) 3³13 A1(7) 7

G2(4) 2⁵13 A1(11) 11

A3(2) 8 ²F4(2)⁰ 2⁶5²

B2(3) 27 ²A2(5) 50

An(q) q^n‡1 1

q 1 Bn(3); (n3) 1

23^{n 1}(3ⁿ 1) Bn(q) q²ⁿ 1

q 1 Cn(q) q²ⁿ 1

q 1 Bn(2) 2^{n 1}(2ⁿ 1) ²An(2);(6jn‡1) 2ⁿ(2^n‡1 1)

3

2A2(q) q³‡1 ²A3(q) (q‡1)(q³‡1)

Dn(q) (qⁿ‡1)(q^{n 1} 1)

q 1 ²An(q);(n4) [q^n‡1 ( 1)^n‡1][qⁿ ( 1)ⁿ] q² 1

2Dn(q) (qⁿ‡1)(q^{n 1} 1)

q 1 ³D4(q) (q‡1)(q⁸‡q⁴‡1) G2(q) q⁶ 1

q 1 E6(q) (q⁴‡1)(q⁹ 1)(q¹² 1) (q³ 1)(q 1)

2B2(q) q²‡1 ²E6(q) (q⁴‡1)(q⁹‡1)(q¹² 1)

(q³‡1)(q 1) F4(q) (q⁴‡1)(q¹² 1)

q 1 ²F4(q) (q‡1)(q³‡1)(q⁶‡1)

2G2(q) q³‡1 E7(q) (q¹⁴ 1)(q⁸ 1)(q¹² 1)

(q⁶ 1)(q⁴ 1)(q 1) Dn(2) 2^{n 1}(2ⁿ 1) E8(q) (q¹⁰‡1)(q¹⁶ 1)(q²⁴ 1)

(q⁶ 1)(q 1)

ther are A2(4) and A3(2) or are Bn(q) and Cn(q) for somen3 and some odd q: So our task of recognizing a simple group G from its Dirichlet polynomial would be nearly completed if we could determine jGj from P_G(s);however, as we mentioned in Section 2, this is not an easy problem, and it can be tackled only with an intensive use of results on the maximal subgroups of Lie groups. However the following remark is useful and easy to prove.

LEMMA 12. Let G be a simple group of Lie type and let B be a Borel subgroup. IfjG:Bj ˆu, then u2y(G):

PROOF. As we noticed in the proof of Theorem 3,m_G(B)ˆ( 1)^lwherel is the Lie rank in the untwisted case and the number ofr-orbits on the nodes of the Dynkin diagram in the twisted case. Now suppose jHj ˆ jBj and m_G(H)6ˆ0;asjG:Hj ˆuis coprime withp;the subgroupHcontains a Sylow p-subgroupPofG; the normalizerB ˆN_G(P) is again a Borel subgroup and is contained in each maximal subgroup of Gwhich containsP; asHis an intersection of maximal subgroups,B H, butjG: Bj ˆ jG:Hj ˆu;hence HˆB; as the Borel subgroups inGare all conjugated and self-normalizing, we conclude thatau(G)ˆ jG:N_G(B)jm_G(B)ˆ( 1)^lu: p The previous lemma tells us that the p⁰-part of po(G) cannot be too different fromjGj_p0;indeedjGj_p0ˆ jG:BkHj;beingHa Cartan subgroup of the Borel subgroup B, hence (jGj=po(G))_p⁰ divides jHj: What is more difficult to understand is how much smaller can (po(G))_p be compared to jGj_p:However, if we already know thatp is the characteristic of the Lie group G;then, by Corollary 4, with the help of the Dirichlet polynomial P_G(s) we may compute the number po(G)ˆl:c:m:(jGj_p;po(G)), and use this number as a good approximation forjGj:For some groups of low rank (A₁(q);²B₂(q);²A₂(q)); we will need a more accurate estimate of po(G):

Before starting the analysis of these particular cases, let us recall some definitions and results concerning Zsigmondy primes. A prime numberuis called aprimitive prime divisorofa^b 1 if it dividesa^b 1 but it does not divide a^e 1 for any integer 1eb 1. It was proved by Zsigmondy [22] that ifaandbare integers greater than 1 and (a;b)6ˆ(2;6);then there exists a primitive prime divisor of a^b 1 except when aˆ2 and b is a Mersenne prime.

LEMMA13. The following hold:

(1) If GˆA₁(q)then q(q 1)=2dividespo(G);

(2) if Gˆ²B2(q)then q 1dividespo(G);

(3) if Gˆ²A2(q)with qˆ3^rand r>1;then(q‡1)²(q 1)divides po(G):

PROOF. (1) IfGˆA₁(q);then looking at the list of its maximal sub- groups (see for example [15], Satz 8.27, pag. 213) one can notice the fol- lowing: if q6ˆ7;9;11; then GˆA₁(q) has a maximal subgroup D iso- morphic to the dihedral group of order 2(q‡1)=(2;q 1) andjMj ˆ jDjfor eachMmaximal subgroup ofGwith index dividingjG:Dj(more precisely eitherMis conjugate toDorqˆ59 andMAlt (5));so ifq6ˆ7;9; 11, then jG:Dj ˆq(q 1)=22~y(G). Finally a direct computation shows that po(G)ˆ jGjwhenG2A₁(7);A₁(9);A₁(11):

(2) IfGˆ²B2(q) thenqˆ2^ewithe3 odd;jGj ˆq²(q²‡1)(q 1) and the maximal subgroups ofGare the following [21]:

(a) B(the parabolic subgroup) withjBj ˆq²(q 1);

(b) XawithjXaj ˆ4(q‡a 

p2q

‡1) anda2 f 1;1g;

(c) D'D_{2(q 1)} withjDj ˆ2(q 1);

(d) ²B2(q0) whereqˆq^a₀andais a prime divisor ofe.

Notice that (q‡ 

p2q

‡1)(q 

p2q

‡1)ˆq²‡1 so there exists a2 f 1;1gsuch thatjXajis divisible by a primitive prime divisoruof 2^4e 1.

A maximal subgroupMhas order divisible byuonly whenMX_a, hence jG:X_aj ˆq²(q a 

p2q

‡1)(q 1)=42~y(G);andq 1 divides po(G):

(3) Forqˆ3^r;r6ˆ1;Gˆ²A2(q)ˆPSU(3;q) has orderq³(q³‡1)(q² 1) and contains the following maximal subgroups [2] (hereZndenotes the cyclic group of ordern):

(a) B[q³]:Z_q² ₁; (b) GU(2;q);

(c) (Zq‡1)²:Sym(3);

(d) Z_q² _q‡1:3;

(e) PSU(3;q₀) withqˆq^a₀ andaprime;

(f) SO(3;q):

Let now ube a primitive prime divisor of 3^6r 1: IfM is a maximal subgroup ofGwith order divisible byuthenMis of the kind described in (d), hencejGj=(3(q² q‡1))ˆq³(q‡1)²(q 1)=32~y(G): p

Now we are ready to start with the proof of our main result.

THEOREM14. Let G1and G2be two simple groups of Lie type defined over fields with the same characteristic; if P_G1(s)ˆP_G2(s), then G₁G₂:

The proof is quite long and will be splitted in consecutive steps. The main ingredient is an analysis performed in [16], which explains how one can recognize a finite simple group of Lie type from its order. For the convenience of the reader we recall some definitions introduced in [16]. Let n>1 be a natural number, thecontributionof a primeptonis the highest power ofpdividingn; the following invariants can be defined:

p(n): is called thedominant primein nand it is the prime number whose contribution tonis maximal.

l(n): is the exponent ofp(n) so thatp(n)^l(n)is the largest power ofp(n) dividingn:

v(n): is the largest order of p(n) modulo a prime divisor p1 of n=p(n)^l(n); such a prime number is called aprominent prime.

c(n): is the largest order ofp(n) modulo a non-prominent prime; one putsc(n)ˆ0 in casenhas no non-prominent prime divisor other thanp(n).

If G is a group of order n than the numbers p(G)ˆp(n); l(G)ˆ

ˆl(n); v(G)ˆv(n) and c(G)ˆc(n) are called the Artin invariants of jGj: In [16] the authors prove that apart from few exceptions, there is a unique simple group of Lie type with given values of the Artin invariants.

Our first task will be to prove that we can obtain the Artin invariants by looking at the Dirichlet polynomial of the group. From now on,Gwill be a finite group of Lie type over a field of characteristic p: In generalp(G) coincides with the characteristicpof our group of Lie type; indeed we have the following (see [16, Theorem 3.3]):

STEP 1. Assume that G is a group of Lie type over a field of char- acteristic p with G6ˆ²A2(3);²A3(2);then p(G)6ˆp if and only if m(G)is a prime-power with m(G)>jGj_p:Moreover one of the following occurs:

(1) m(G)is a 2-power, in which case pˆm(G) 1is a Mersenne prime and GˆA₁(p);

(2) m(G) is a Fermat prime, in which case pˆ2 and Gˆ

ˆA1(m(G) 1);

(3) m(G)ˆ9;in which case pˆ2and GˆA₁(8):

This has the following consequence:

STEP2. Let G be a simple group of Lie type defined over a field F; if we know that the characteristic of F is p;then we may determine from the Dirichlet polynomial P_G(s)whether p(G)ˆp or not, and when p(G)6ˆp we may determine G up to isomorphism.

PROOF. First notice thatGˆ²A2(3) is the unique simple group of Lie type withm(G)ˆ27;on the other handP(²A3(2))ˆ28 andGˆ²A3(2) is the unique group of Lie type withm(G)ˆ28 and characteristic 2. First notice thatGˆ²A₃(2)'B₂(3) is the unique group of Lie type withm(G)ˆ27 (see for example the list of primitive groups of small degree in [11]) so we may recognizeGfrom its Dirichlet polynomial. Next let us considerGˆ²A₂(3), here we getm(G)ˆ28 andp(G)ˆ26ˆ3ˆp; ifHis a group of Lie type with m(H)ˆ28, thenH2 f²A2(3);A1(27);B3(2)g. It can be checked (see Lemma 13 and [5]) that 13272~y(A₁(27)) and 5242~y(B₃(2)) but neither of these numbers dividesjGjso ifH6'G, thenP_G(s)6ˆP_H(s).

When G62 f²A₃(2);²A₂(3)g we proceed as follows: we determinem(G) andjGj_p with the help of the Dirichlet polynomialPG(s);ifm(G) is not a prime-power orm(G) jGj_p);thenpˆp(G);otherwise we are in one of the three cases described in step 1 and in each of them the knowledge ofm(G)

suffices to determineGup to isomorphism. p

So our theorem is proved whenp(G)6ˆp;and for the rest of the proof our attention will be restricted on groups of Lie type satisfyingp(G)ˆp:

Clearly, in this case we can obtain alsol(G) from the knowledge ofP_G(s) as p^l(G) ˆ jGj_p:Now we start to discuss the other two Artin invariants,v(G) andc(G). Let us first recall other results from [16]. The order ofGˆL(q) has a standard factorization:

jL(q)j ˆ1 dq^hP(q);

whered,handP(q) are given in [16, Table L1]. In particular (see [16]) this order has thecyclotomic factorization in terms of p:

jL(q)j ˆ1 dp^lY

m

Fm(p)^em;

where F_m(x) is the m-th cyclotomic polynomial. Let a(G) be the largest value ofmfor whichem 6ˆ0 and defineb(G) as the next largest value ofm for whichem6ˆ0;as it is explained in [16], a consequence of Zsigmondy's Theorem is that, apart from the few exceptional cases listed in [16] Lemma 4.6,a(G) andb(G) coincides withv(G) andc(G) (and the precise values are reported in [16] Table A.1).

STEP3. If p(G)ˆp;then(v(G);c(G))ˆ(v(po(G));c(po(G))):

PROOF. As we noticed after the proof of Lemma 12,jGj=po(G) divides the order of a Cartan subgroupHofG:For most cases, in order to prove our

statement it will suffice to check that the prime numbers dividingjHjare not relevant when one wants to computev(G) andc(G), which is equivalent to verify the following condition:

() there exist at least two primes u1and u2 which dividejGj=jHj such that the following holds: for any prime u dividingjHj;the order of p modulo u is not grater than the order of p modulo u1

and u₂.

So before starting with a case by case analysis we recall the value ofjHj for the different classes of groups of Lie type. IfGˆL_n(q) is of untwisted type, thenjHj ˆ(q 1)ⁿ=d(see Section 8.6 in [4]); ifGˆⁱL_n(q) is a twisted Lie group and all the roots have the same length, thenjHj ˆQ

J (q^jJj 1)=d;

whereJruns on the set of ther-orbits on the nodes of the Dynkin diagram (see Section 14.1 in [4]); ifG2 f²B2(q);²G2(q);²F4(q)g(i.e. whenGhas roots of different length), thenjHj ˆ(q 1)^e;withe2 f1;2g. Now we start with the analysis of the different possibilities; according to [16] Lemma 4.6, there are three cases:

a) (v(G);c(G))ˆ(a(G);b(G)):

In this case the values of (v(G);c(G)) are given in [16] Table A.1.

Assume that G is a Lie group over the field F_q with qˆp^r; if G is of untwisted type, then any primeuwhich divides jHj, also dividesq 1, hencephas order at mostrmodulou; this means that condition () is certainly verified when v(G)>c(G)>r and, by [16] Table A.1, this is true with the only exception ofGˆA1(q);in this last case we can easily conclude with the help of Lemma 13. Now assume that Gis a twisted Lie group. First consider the cases when all the roots have the same length. If uis a prime divisor ofjHj, then the order of pmoduloudi- vides rs, with sˆ3 when Gˆ³D4(q); and sˆ2 otherwise. If G6ˆ²A2(q), then v(G)>c(G)>rs and () is satisfied. If Gˆ²A2(q);

then (v(G);c(G))ˆ(6r;2r);this information may be recovered from the index of the Borel subgroup B as jG:Bj ˆq³‡1 is divisible by F_6r(p)F_2r(p): If G2 f²B₂(q);²G₂(q);²F₂(q)g, then p has order at most r modulo any prime divisor of jHj ˆq 1 and condition () is certainly satisfied except forGˆ²B2(2^r) with r 1 mod 6:In this last case one can apply Lemma 13 (2).

b)pˆ2;a(G)ˆ6 orb(G)ˆ6:

The pairs (v(G);c(G)) and (a(G);b(G)) does not coincide when pˆ2 and either a(G)ˆ6 or b(G)ˆ6; there are precisely 18 groups of Lie type in this situation, and the values (v(G);c(G)) for them are given in

[16, Table A.2(a)]; for most of them a direct and easy computation shows that the exist at least two primes u1 and u2 dividing the index of the Borel subgroup and such that p has order v(G) modulo u1 and c(G) modulou₂; only two of these groups require more attention. The first of them isGˆA₂(8): in this casejGj ˆ2⁹3²7²73;v(G)ˆ9 is the order of 2 modulo 73 and c(G)ˆ3 is the order of 2 modulo 7; but the Borel subgroup B has index 3²73, not divisible by 7, so we need to check more carefully whether 7 divides po(G):this can be deduced by looking at the indices of the maximal subgroups, indeed it turns out that 2⁹37²2~y(G):A similar situation occurs whenGˆ²A₂(8): in this case jGj ˆ2⁹3⁴719;and v(G)ˆ18;c(G)ˆ3 are the orders of 2 modulo 19 and 7; 7 does not divide the index of the Borel subgroup but 2⁸7192~y(G):

c) p is Mersenne, b(G)ˆ2; c(G)ˆ1 andGˆA1(p²);A2(p); B2(p) or

2A₂(p):

This case does not give us particular problems; same argument which has been applied in case (a) tells us that v(G)ˆa(G) is the order of p modulo a prime divisor of the index of the Borel subgroupB(see [16, Table A.3]). MoreoverjG:Bjis an even number andc(G)ˆ1 is the order ofp

modulo 2. p

In [16] the authors prove that the are only few pairs of non-isomorphic simple groups of Lie type with the same Artin invariants (the possibilities are listed in [16], Table 5.2). In particular we have:

STEP 4. Suppose that G₁ and G₂ are two non isomorphic simple groups of Lie type with the same Artin invariants; if m(G₁)ˆm(G₂)then one of the following occurs:

(1) G₁ˆ²A₂(q);G₂ˆ²G₂(q);

(2) G₁ˆ²A₂(q²);G₂ˆ²B₂(q³);

(3) G1ˆA1(2⁶); G2ˆ²A2(4);

(4) G1ˆBn(q);G2ˆCn(q)with q odd and n3:

STEP5. P²_A2(q)(s)6ˆP²_G2(q)(q):

PROOF. Assume, by contradiction, that P2A2(q)(s)ˆP2G2(q)(s): This would imply po(²A2(q))ˆpo(²G2(q));since, by Lemma 13, (q‡1)² divides po(²A2(q)); we would get that (q‡1)² divides j²G2(q)j ˆq³(q³‡1)(q 1);

which is false. p

STEP6. P²_A2(q²₎(s)6ˆP²_B2(q³₎(s):

PROOF. Assume, by contradiction, that P²_A2(q²₎(s)ˆP²_B2(q³₎(s): This would imply po(²B2(q³))ˆpo(²A2(q²));since, by Lemma 13, q³ 1 divides po(²B2(q³)); we would get thatq³ 1 divides j²A2(q²)j ˆq⁶(q⁶‡1)(q⁴ 1);

which is false. p

STEP7. P_A1(2⁶₎(s)6ˆP²_A2(4)(s).

PROOF. Assume by contradiction that P_A1(2⁶₎(s)ˆP²_A2(4)(s). Thus po(A₁(2⁶))ˆpo(²A₂(4)). By Lemma 13 we get that 72po(A₁(2⁶)), but j²A₂(4)j ˆ2⁶35²13 is not divisible by 7. p

STEP8. If q is odd and n3:then P_Bn(q)(s)6ˆP_Cn(q)(s):

PROOF. Recall that jBn(q)j ˆ jCn(q)j ˆqⁿ² Q

1in(q²ⁱ 1)

=2: As-

sume that qˆp^r with p an odd prime, and let p be the union of the sets p₁ and p₂, where p₁ is the set of the primitive prime divisors of p^2nr 1 and p2 the set of the primitive prime divisors of p^{(2n 2)r} 1:

Our aim is to prove that P^(p)_Bn(q)(s)6ˆP^(p)_Cn(q)(s): We will apply a theorem of Feit ([12, Theorem A]) which asserts that if (2n;q)2 f(6;= 3);(6;5)g;

then there exists u2p₁ such that either u>2nr‡1 or u² divides p^2nr 1ˆq²ⁿ 1: By definition, the non trivial contributions to the computation of the Dirichlet polynomial P^(p)_G(s) come only from the subgroups containing a Sylow u-subgroup of G for each prime u2p:

The maximal subgroups of Cn(q)PSp (2n;q) whose order is divisible by at least a prime u₁2p₁ and a prime u₂2p₂ are described in [13]

and [17, Table 2.5]. In particular we deduce that if C_n(q)PSp (2n;q) contains a maximal subgroup M with this property, then (2n;q)2 f(6;= 3);(6;5)g, rˆ1; u1ˆ2n‡1 and jMj is not divisible byu²₁; as a consequence we get that uˆu1 and u² divides jCn(q)j whereas u does not dividejMj. This implies that there is no maximal subgroup of C_n(q) containing a Sylow u-subgroup for each u2p; hence P^(p)_Cn(q)(s)ˆ1: The situation is different for B_n(q)V_2n‡1(q): Indeed if W is a non-singular 1-dimensional subspace of the orthogonal space V F^2n‡1_q with the property that W^? has typeO_2n; then the stabilizer MˆStab_V2n‡1(q)(W) is a maximal subgroup of V_2n‡1(q) with jV_2n‡1(q):Mj ˆqⁿ(qⁿ 1)=2, a p⁰-number. This implies P^(p)_Bn(q)(s)6ˆ1ˆ

ˆP^(p)_Cn(q)(s): p

6. An example.

The following is the Dirichlet polynomial associated with a finite simple group G; we want to use the results established in the paper in order to identifyGup to isomorphism.

P_G(s)(S)ˆ1 62

31^s‡ 186

(2331)^s‡ 775 (5²31)^s

3100 (2²5²31)^s 3875

(5³31)^s

4000 (2⁵5³)^s

4650 (235²31)^s

‡ 11625

(35³31)^s‡ 15500

(2²5³31)^s‡ 18600 (2³35²31)^s

‡ 31000 (2³5³31)^s

186000

(2³35³31)^s‡ 124000 (2⁵5³31)^s First we notice thatm(G)ˆ31;asa₃₁(G)ˆ 62, from Theorem 1 we de- duce that Gis not of alternating type; moreover by Table 1, there is no sporadic simple groupHwithm(H)ˆ31;henceGis a simple group of Lie type. Now we computeP^(p)_G (0) for any primep2p(po(G))ˆ f2;3;5;31g;we get the following values:

p 2 3 5 31

P^p†_G …0† 2⁴23² 3311723 5³ 33143

Only forpˆ5 the numberjP^(p)_G(0)jis ap-power, hence by Theorem 3 the Lie group G is defined over a field of characteristic 5, and 5³ is the order of a Sylow 5-subgroup ofG:Moreoverm(G)ˆ31 jGj₅ˆ5³so by the argument in Step 1 of the proof of Theorem 14,p(G)ˆ5 andl(G)ˆ3:

The orders of 5 modulo 2,3,31 are, respectively, 1,2,3 so v(G)ˆ3 and c(G)ˆ2: G has Artin invarians (p(G);(G);v(G);c(G))ˆ(5;3;3;2); this allows us to concludeGA2(5)ˆ(3;5):

REFERENCES

[1] E. ARTIN,The orders of the classical simple groups. Comm. Pure Appl. Math., 8(1955), pp. 455±472.

[2] D. M. BLOOM,The subgroups of PSL(3;q)for odd q.Trans. Amer. Math. Soc., 127(1967), pp. 150±178.

[3] K. S. BROWN,The coset poset and probabilistic zeta function of a finite group.

J. Algebra 225,2(2000), pp. 989±1012.

[4] R. W. CARTER,Simple groups of Lie type.Wiley Classics Library. John Wiley

& Sons Inc., New York, 1989. Reprint of the 1972 original, A Wiley- Interscience Publication.

[5] J. H. CONWAY- R. T. CURTIS- S. P. NORTON- R. A. PARKER- R. A. WILSON, Atlas of finite groups. Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray.

[6] E. DAMIAN, - A. LUCCHINI,The Dirichlet polynomial of a finite group and the subgroups of prime power index.InAdvances in group theory 2002. Aracne, Rome, 2003, pp. 209±221.

[7] E. DAMIAN - A. LUCCHINI, Recognizing the alternating groups from their probabilistic zeta function. Glasgow Math. J.,46(2004), pp. 595±599.

[8] E. DAMIAN, - A. LUCCHINI,The probabilistic zeta function of finite simple groups. J. Algebra, submitted.

[9] E. DAMIAN- A. LUCCHINI- F . MORINI,Some properties of the probabilistic zeta function on finite simple groups. Pacific J. Math.,215, 1 (2004), pp. 3±14.

[10] E. DETOMI- A. LUCCHINI,Recognizing soluble groups from their probabilistic zeta functions. Bull. London Math. Soc.,35, 5 (2003), pp. 659±664.

[11] J. D. DIXON- B. MORTIMER,Permutation groups, vol.163ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1996.

[12] W. FEIT,On large Zsigmondy primes. Proc. Amer. Math. Soc.,102, 1 (1988), pp. 29±36.

[13] R. GURALNICK- T. PENTTILA- C. E. PRAEGER- J. SAXL,Linear groups with orders having certain large prime divisors. Proc. London Math. Soc. (3)78, 1 (1999), pp. 167±214.

[14] P. HALL,The eulerian functions of a group. Quart. J. Math.,7(1936), pp. 134-151.

[15] B. HUPPERT, Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer-Verlag, Berlin, 1967.

[16] W. KIMMERLE- R. LYONS- R. SANDLING- D. N. TEAGUE,Composition factors from the group ring and Artin's theorem on orders of simple groups. Proc.

London Math. Soc. (3)60, 1 (1990), pp. 89±122.

[17] M. W. LIEBECK- C. E. PRAEGER- J. SAXL,The maximal factorizations of the finite simple groups and their automorphism groups. Mem. Amer. Math.

Soc.,86, (1990), p. 432.

[18] D. QUILLEN,Homotopy properties of the poset of nontrivial p-subgroups of a group. Adv. in Math.,28, 2 (1978), pp. 101±128.

[19] L. SOLOMON, The Steinberg character of a finite group with BN-pair. In Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968). Benjamin, New York, 1969, pp. 213±221.

[20] R. P. STANLEY,Enumerative combinatorics. Vol. 1, vol. 49 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997.

[21] M. SUZUKI, A class of doubly transitive permutation groups. In Proc.

Internat. Congr. Mathematicians (Stockholm, 1962). Inst. Mittag-Leffler, Djursholm, 1963, pp. 285±287.

[22] K. ZSIGMONDY,Zur Theorie der Potenzreste. Monatsh. Math. Phys.,3(1892), pp. 265±284.

Manoscritto pervenuto in redazione il 7 settembre 2005.