JOURNAL OF ALGEBRA 194, 79᎐112Ž. 1997 ARTICLE NO. JA966980
Short Presentations for Finite Groups
L. Babai*†
Uni¨ersity of Chicago, Computer Science Department, 1100 E. 58th St., Chicago, Illinois 60637 and Mathematical Institute of the Hungarian Academy of Science, Budapest, Hungary
A. J. Goodman*
Uni¨ersity of Missouri-Rolla, Mathematics and Statistics Department, Rolla, Missouri 65409
W. M. Kantor*
Uni¨ersity of Oregon, Mathematics Department, Eugene, Oregon 97403
E. M. Luks*
Uni¨ersity of Oregon, Computer and Information Science Department, Eugene, Oregon 97403
and
P. P. Palfy´ ‡
Mathematical Institute of the Hungarian Academy of Science, Budapest, P. O. Box 127, H-1364 Hungary
Communicated by Robert Steinberg
Received February 8, 1996
We conjecture that every finite group G has a short presentationŽ in terms of generators and relations. in the sense that the total length of the relations is Žlog< * Research supported in part by NSF grants. † E-mail: [email protected], [email protected], [email protected], luks@cs. uoregon.edu, [email protected]. ‡ Research supported in part by Hungarian Research Fund Grant OTKA-1903. 79 0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 80 BABAI ET AL. We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name of G, assuming in the case of Lie type simple groups over GFŽ p m .Ž.that an irreducible polynomial f of degree m over GF p and a primitive root of GFŽ pm . are given. We verify thisŽ. stronger conjecture for all finite simple groups except for the three families of rank 1 twisted groups: we do not handle the unitary groups Ž.2Ž. Ž. 2 Ž. Ž. PSU 3, q s Aq22, the Suzuki groups Sz q s Bq, and the Ree groups Rq 2Ž. sGq2 . In particular, all finite groups G without composition factors of these types 3 ha¨e presentations of length OŽŽlog< CONTENTS 1. Introduction. 2. Groups other than Lie type. 3. Chevalley groups. 4. Presentations for the universal Chevalley groups of rank G 2. 4.1. The Steinberg and Curtis᎐Steinberg᎐Tits presentations. 4.2. The reduced set. 4.3. Length of the presentation. 5. Decreasing the length and eliminating the center. 5.1. OŽlog< 1. INTRODUCTION A presentation of a group is a description by generators and relations which defines the group; we allow relations written as equations using powers of the generators. For general background and a number of specific presentations, we refer to Coxeter and Moserwx CM72 . SHORT PRESENTATIONS 81 We define the length of a presentation to be the total number of characters required to write all the relations. We write exponents in binary and include the number of digits in this count. We shall only be interested in the asymptotic order of magnitude of presentation lengths, so it makes no difference whether or not we count parentheses, equal signs, and minus signs; for definiteness, we shall not count them. We use log to denote base 2 logarithms. Each generator will count as a single symbol. m EXAMPLE 1.1. The presentation ² x : x s 1: of the cyclic group of order m has length log m q OŽ.Ž1 since it takes at most 1 q log m digits to write the exponent m in binary.Ž. . The presentation of PSL 2, p given in Remark 7.2 has length OŽ.log p . Remark 1.2. A natural alternative to counting each generator as a single symbol is to assume that the generators are indexed symbols x1,..., xk and to count the digits in these subscripts as part of the length. However, this would only increase the length of a presentation by a factor of log k, an insignificant factor from our point of view since log k is OŽlog log< THEOREM 1.4. If Conjecture 1 holds for all finite simple groups with some C 1 CG2, then e¨ery finite group G has a presentation of length OŽlog q < DEFINITION 1.1. Assume that ‘‘standard names’’ are associated with the members of a class F of finite groups. We say that F has efficiently ¨erifiable presentations if there exists a constant C and a deterministic multitape Turing machine M accepting triples Ž.G, P, W of strings such that Ž.i whenever G is the standard name of some member of F, there exist P and W such that P is a presentation of G and M accepts ŽW is the ‘‘witness’’ of correctness of the presentation. ; Ž.ii if G is a standard name of some member of F and P is not a presentation of G, then M rejectsŽ. no ‘‘false witness’’ exists ; Ž.iii if G is the standard name of some member of F, then M halts in OŽlogC < m qsp. An explicit representation of the field GFŽ. q , i.e., an irreducible polynomial of degree m over GFŽ. p , is likely to be needed for the computation. No deterministic algorithm, running inŽ. log q OŽ1.time, is known at present to produce such a polynomial except under the extended Riemann hypothesisŽ Adleman and Lenstrawx AL86.Ž . Randomized algo- rithms do exist for this purpose, and the verification that a given polyno- mial is irreducible can be performed deterministically inŽ. log q OŽ1.time wxBer70, CZ81, Rab80 .. We shall not be concerned with this difficulty and get around it by the following convention. CONVENTION Ž.) . In case of a group G of Lie type over the field GFŽ p m., we require that the standard name of G include an irreducible polynomial of degree m over GFŽ. p . This convention seems quite natural and amounts to saying that the underlying field itself rather than its order has to be specified as part of the name of the group. It is less natural, but it seems necessary in some cases, that we have, in addition, a primitive rootŽ generator of the multi- plicative group.Ž. of GF q at hand. Without this, we presently are not able to turn a presentation of SLŽ. n, q into one of PSL Ž.Ž n, q . However, this step and its analogues for the other Lie type groups are the only places where we will need the primitive root.. Unfortunately, no deterministic algorithm is known to find a primitive root of GFŽ. q in time Ž log q .OŽ1., even for the case m s 1 Ž.q s p a prime . Given these additional tools, however, we believe that presentations can be computed efficiently from the names of the finite simple groups.Ž For twisted groups the relevant field may be GFŽ q 2 .Žor GF q 3.in terms of the q given in the standard name, and we assume that an irreducible polynomial and primitive root are given for this larger field.. Conjecture 3Ž. Uniform Short Presentation Conjecture . There exists a constant C and an algorithm which computes a presentation of each finite simple group G in time OŽlogC < More precisely, we conjecture that C s 2 suffices. ŽŽ.Ž.Ž.Given f as in b1 , we represent GF q as GF pwx x rŽ.f ; and Ž b2 . is understood to conform to this representation, i.e., the primitive root is given by a polynomial in GFŽ. pwx x whose class mod Ž.f is the primitive root in GFŽ. pwx x rŽ..f . 84 BABAI ET AL. Each of the three conjectures is stronger than its predecessor: Conjec- ture 3 implies Conjecture 2 even without assuming Ž.) , because the additional inputŽ. b1 and Ž. b2 may be contained in the witness W Žthe machine in Definition 1.1 may use its inputs G and W together to compute a presentation which it then compares with the given presentation P.. Thus it suffices to consider Conjecture 3. For clarity, the proofs will be worded to stress the validity of Conjecture 1; the presentations obtained will be so explicit that the algorithmic requirements in the other conjec- tures will evidently be met. THEOREM 1.6. The uniform short presentation conjecture holds, with C s 2, for all finite simple groups, with the possible exception of the rank 1 twisted groups of Lie type. The complexity theoretic application are of a qualitative nature. First stated inwx BS84 , Conjecture 2 is known to imply that certain problems for matrix groups G over finite fields belong to the complexity class NPŽ cf. wxGJ79 for information regarding this complexity class. , where a group is given by a list of generating matrices. These problems include nonmember- ship of a given matrix in G, verification of the order of G wxBS84 , and isomorphism of two such groups G, H Žcf.wx Ba92, Propositions 4.9 and 4.10 for this result of Luks.Ž . The proof that the membership problem belongs to NP is elementary; it is an immediate consequence of the reachability lemmawx BS84 , stated in Section 8 of this paper.. For a list of further consequences of Conjecture 2, seewx Ba92, Corollary 12.1 . Conjecture 3 is expected to be useful in efficient algorithms for matrix groupsŽ cf.wx Ba97. . The algorithmic parts of the present paper deal only with simple groups. We do not address algorithmic questions for the nonsimple groups in Conjecture 1, nor the question of finding the standard name of a simple group given in some other way. 2. GROUPS OTHER THAN LIE TYPE As customary, for two sequences of positive real numbers Ä4annand Ä4b , Ž. Ž. we use the expression anns ⌰ b to denote that a nns Ob and b ns OaŽ.n . First we note that the sporadic groups do not matter for us. Even if we use their full Cayley tables for presentations, this will not influence the validity of the conjectures. Example 1.1 takes care of cyclic groups G: their natural presentations have length ⌰Žlog< A number of suitable presentations for the alternating groups G s An 2 are given inwx CM72, Sect. 6.3 . All of them have length ⌰Žn .s 2 ⌰ŽŽlog< Generators: xiiŽ.s1,...,ny2, 33иии 2 Relations: x1s s xny2s Ž.xxij s11 ŽFi-jFny2. . 3. CHEVALLEY GROUPS The remaining finite simple groups are of Lie type. They fall into two categories: the normal and the twisted types. We shall refer to the former as Che¨alley groups. Presentations for the groups of Lie type were first given by Steinbergwx Ste 62, Ste 81Ž. cf. Sections 4 and 6 below . In Sections 3 to 5 we consider the case of Chevalley groups, postponing the discussion of the twisted types until Section 6. Each Chevalley group G is associated with a finite field of order q and a parameter N Žthe number of ‘‘positive roots’’. . Steinberg uses 2 Nq generators. The Curtis᎐Stein- berg᎐Tits presentationwx Cur65 reduces this number somewhat, but its length is still proportional to q, not log q Ž.cf. Section 4 . This is not satisfactory from our point of view since log< Let L stand for one of the letters A, B, C, D, E, F, G. Let Ln denote the corresponding complex simple Lie algebra of rank n.Ž For type A, Ž. nG1; for types B and C, n G 2 here B22s C ; for type D, n G 4; for type E,6FnF8; for type F, n s 4; and for type G, n s 2.. Types A, B, C, and D are called classical, the others exceptional. Let LnŽ.q denote the universal Chevalley group over the field of order m Ž. qsppprime , corresponding to Ln. The number n is the rank of the group. Remark 3.1. In this classification, the classical simple groups appear as follows. The left column indicates universal Chevalley groups in Lie 86 BABAI ET AL. notation; the middle column their central quotient, a classical simple group; the right column their classical name. AnŽ. q PSL Ž n q 1, q . linear BqnŽ. P⍀ Ž2nq1, q . orthogonal CnŽ. q PSp Ž2n, q . symplectic q DqnŽ. P⍀ Ž2n,q . orthogonal There is another class of simple orthogonal groups of even dimension, P⍀yŽ.2n,q. They, along with the unitary groups, correspond to twisted types: 2 AnŽ. q PSU Ž n q 1, q . unitary 2 y DqnŽ. P⍀ Ž2n,q . orthogonal where, however, n is not the rank of the twisted group. Remark 3.2. In almost all cases the universal Chevalley group Gˆ is the uni¨ersal central extension of the simple group G, but there are a few cases where Gˆ has nontrivial Schur multiplierwx Ste81 . For convenience, we treat these finitely many exceptions as sporadic groupsŽ they cannot affect our asymptotic results. and exclude them in the following sections, because the Curtis᎐Steinberg᎐Tits presentation is knownwx Cur65 to describe a central extension of Gˆˆˆ, which therefore must be G if G has trivial Schur multiplier. If the rank is 2, then the Curtis᎐Steinberg᎐Tits presentation is the same as the Steinberg presentation for Gˆ, so we only need to exclude the cases of rank G 3 where Gˆ has nontrivial Schur multiplier, namely Ž. Ž. Ž. Ž. Ž. Ž. A332,B 2 (C 3342,B 3,D 2 , and F 42. The same remark applies to the central extensions of twisted groups 2 2 considered in Section 6, where we exclude A56Ž.2 and E Ž.2. 4. PRESENTATIONS FOR THE UNIVERSAL CHEVALLEY GROUPS OF RANK G 2 In order to simplify the notation, in Sections 4 and 5 we will change notation and now let G stand for the universal Chevalley group LnŽ.q . Let q ⌽ denote the set of roots of the simple Lie algebra Ln. Let ⌽ be a q positive system of roots; N s <⌽ << N s ÝŽ.diy 11 Ž. is1 SHORT PRESENTATIONS 87 and Nd < 2 N 2N n 3N q -< 4.1. The Steinberg and Curtis᎐Steinberg᎐Tits Presentations Ä Ž. Ž .4 Steinberg introduces the set xt␣ :␣g⌽,tgGF q of 2 Nq genera- tors for G and shows the following set of relations to be sufficient to define G in the case n G 2wx Ste67, Theorem 9, p. 72 . Ž.A xt␣ Žqu .sxtxu␣␣ Ž. Ž.Ž. t,ugGF Ž. q , ␣ g ⌽ ; ij Ž.B xt␣ Ž.,xu Ž.sŁxCtui␣qjŽ.i,j,␣, i,j)0 for ␣ ,  g ⌽, ␣ /",t,ugGFŽ. q . Here wxg, h denotes the commutator ghghy1y1; the product is taken over those values i, j ) 0 for which i␣ q j  is a root; therefore the product never has more than four terms and the relevant values of i, j are not greater than 3; the coefficients Ci, j, ␣ ,  thus form a finite listŽ independent of n and q. for given ␣, . These coefficients are integers of absolute value F 3Ž. most often "1 and are explicitly computed inwx Ste67, Sect. 10 . In all cases except Gq2Ž., the terms in the product in Ž. B commute and thus their order does not matter. The case Gq2Ž.is described in detail in wxSte67, Sect. 10 . A subset of the above generators and relations will suffice for a presentation of G Ž.see Remark 3.2 , namely the Curtis᎐Steinberg᎐Tits presentationwx Cur65 , which can be described as follows. For 1 F i - j F n, let ⌽ij denote the rank 2 subsystem of ⌽ spanned by the ith and jth fundamental rootsŽ we are assuming a fixed choice of a positive system of roots ⌽q, or equivalently a fixed base consisting of n . ÄŽ. fundamental roots . Let ⌿ s D i- jij⌽ , let F ijs ␣,  g ⌽ ij= ⌽ ij: 4 Ž ␣/" , and let F s D i- jijF . Note that ⌿ is a subset of ⌽ and F is a set of pairs of roots from ⌿.. Then use only those of Steinberg’s generators Ž. xt␣ for which ␣ g ⌿; and use only those of the above relations of type Ž.A for which ␣ g ⌿ and those of type Ž. B for which the pair Ž␣,  .is in F. In other words, the Curtis᎐Steinberg᎐Tits presentation involves only those relations that come from certain rank 2 subgroups of G Žnamely the 88 BABAI ET AL. ²Ž. Ž.: rank 2 subgroups Gij s xt␣ :␣g⌽ij,tgGF q corresponding to pairs of fundamental roots, i.e., edges or nonedges in the Dynkin diagram. . XAMPLE Ž. Ž. E 4.1wx Ste67, p. 72 . For n G 3, the group SL n, q s Aqny1 Ž.Ž Ž .. is defined in terms of the generators xtij 1Fi,jFn,i/j,tgGF q by the following relations: xtijŽ.qusxtxu ij Ž.Ž. ij , xtijŽ.,xu jk Ž .sxtu ik Ž . if i, j, k are different, xtijŽ.,xu kl Ž . s1ifj/k,i/l. This presentation encodes the relations among the usual matrices I q tEij, where Eij is the matrix whose Ž.i, j entry is 1 with all other entries 0. The Curtis᎐Steinberg᎐Tits presentation of SLŽ. n, q uses only the gen- Ž. erators xtij with < 4.2. The Reduced Set We further reduce the Curtis᎐Steinberg᎐Tits presentation as follows. Ä4 Ž. Ž. Let B s b1,...,bm be a basis of GF q over the prime field GF p . Use Ž.Ž . <<⌿mgenerators yb␣ ␣g⌿,1FFm, and define Ž.A0 k yt␣Ž.[ŁÝyb␣ Ž .whenever t s kb Ž0Fk -p .. Ž.Ž We define a group G0, generated by the symbols yb␣ ␣g⌿,1FF m., subject to the following set of relations. p Ž.A1 yb␣ Ž. s1 Ž␣g⌿,1FFm .; Ž.A2 yb␣ Ž.␣,ybŽ.s1 Ž␣g⌿,1F-Fm .; ij Ž.B0 yb␣ Ž.,ybŽ.sŁyCbbi␣qjŽ.i,j,␣, i,j)0 Ž.Ž.␣,gF,1F,Fm . SHORT PRESENTATIONS 89 ŽŽ.Note: The right-hand side of B0 must be expanded, using the definition Ž. Ž.Ž A0 , into an expression involving the generators yb␣ ␣g⌿,1F Fm... THEOREM 4.2. G0 ( G: each uni¨ersal Che¨alley group of rank n G 2 is defined by a presentation using just the relations Ž.Ž.A1 , A2 , and Ž.B0 . Ž . Ž. Ž. Ž Proof. Using A0 , the correspondence yt␣ ¬xt␣ ␣g⌿,tg GFŽ.. q defines a homomorphism from G0 onto G. We have to show it is one-to-one. It suffices to show that is one-to-one on each of the rank 2 subgroups corresponding to a pair of fundamental roots, because every relation in the Curtis᎐Steinberg᎐Tits presentation occurs inside one of these rank 2 subgroups. More precisely, we proceed as follows: q Consider one of the rank 2 subsystems ⌽ij. Let ⌽ ij be an arbitrary positive system of roots in ⌽ij Žnot necessarily the one induced by our q q fixed positive system ⌽ .Ž. Let G0 ⌽ij.be the subgroup of G0 generated Ä Ž. q 4 by the set S s yb␣ :␣g⌽ij ,1FFm. q Claim 4.3. is one-to-one on G0Ž⌽ij.. q Proof. Let ␣ijand ␣ be the two fundamental roots in ⌽ ij. Then every q member of ⌽ijis a unique linear combination of ␣ iand ␣ j with nonnegative integer coefficientswx Ste67, Appendix I, Proposition 9 . Order q ⌽ij in increasing order of the sum of these coefficients and extend this ordering to S. Then, byŽ. A2 and Ž.Ž B0 with Ž.. A0 together with Ž. A1 , it q q <⌽ ij< follows readily that G0Ž⌽ij. is a p-group of order at most q . q On the other hand, the image of G0Ž⌽ij.under is a Sylow p-subgroup of the rank 2 subgroup of G corresponding to ⌽ij ŽwSte67, corollary after Lemma 54, p. 132; cf. Lemma 21, p. 31x , applied to the rank 2 Chevalley q <⌽ ij< group Gij.ŽŽ..and thus its order is q cf. 2 . This proves Claim 4.3. Ž. Ž . Now consider any relation B for a pair ␣,  g Fij. We claim that the Ž. Ž . Ł Ž ij. corresponding relation wyt␣ ,yu xs i, j) 0 yCtui␣qj  i, j, ␣ ,  holds in q G0. Indeed, since ␣ /", there exists a positive system ⌽ijin ⌽ ij such q that ␣,  g ⌽ij . By Claim 4.3, any relation involving only these positive q roots is satisfied simultaneously in G0Ž⌽ij. and its image in G. Ž. Ž . Thus, we have shown that all relations B for ␣,  g F hold in G0. Ž. q Similarly, each relation A for ␣ g ⌿ occurs in some subsystem ⌽ij Žalternatively, these relations also follow more directly from the relations Ž.Ž.A1 , A2 , and Ž.. A0 . Thus, we have shown that all relations of the Curtis᎐Steinberg᎐Tits presentation for G also hold in G0 , completing the proof of Theorem 4.2. 90 BABAI ET AL. 4.3. Length of the Presentation The presentationŽ.Ž.Ž. A1 , A2 , B0 involves <<⌿ m generators. We show 2 that <<⌿ is OnŽ.Žas opposed to <<⌽ s 2 N which is ⌰Žn ... This can be proved as follows. First observe that any rank 2 root systemŽ in particular any one of the subsystems ⌽ij.Žcontains a bounded number of roots at most 12.Ž. . Next observe that there are only On edges in the Dynkin diagram for any root system ⌽ of rank n, i.e., only OnŽ.‘‘nontrivial’’ subsystems ⌽ij if we call a subsystem of type A11= A ‘‘trivial.’’ Then observe that the ‘‘trivial’’ subsystems ⌽ij corresponding to nonedges in the 2 Dynkin diagramŽŽ i.e., the On.different subsystems of type A11= A , corresponding to pairs of root subgroups which centralize each other. Ä4 contribute nothing new to the set ⌿ s D i- jij⌽ since ⌽ ijs "␣ i, " ␣ j already occurs in the union of the ‘‘nontrivial’’ subsystems. Therefore <<⌿ is OnŽ.. 2 Also observe that < COROLLARY 4.4. Each uni¨ersal Che¨alley group G of rank n G 2 Ž gi¨en by its standard name, using Ž..) has an explicit presentation of length 2 2 3 OmŽ.Ž.Ž.log< 5. DECREASING THE LENGTH AND ELIMINATING THE CENTER In this section we let denote a generator of GFŽ. q over GF Ž. p , and choose our basis B of GFŽ. q over GF Ž. p to consist of the first m powers SHORT PRESENTATIONS 91 y1 Ž. Ž. of . Thus we let b s 1 F F m , and our generators yb␣will y1 correspond to x␣Ž .. 5.1. OŽlog< For ␣ g ⌿ and t g GFŽ. q , t / 0, followingwx Ste67, p. 27 , we let y1 wt␣Ž.syty␣ Ž.y␣␣ Žyt .yt Ž. Ž.4 and y1 ht␣Ž.swtw␣␣ Ž. Ž.1. Ž. 5 ŽŽ.Ž..Here again we implicitly use A0 to expand yt␣ , etc. Steinberg proves that htu␣Ž.shthu␣␣ Ž.Ž.Ž.␣g⌿,t,ugGF Ž. q y Ä406 Ž. follows from his relationswx Ste67, Theorem 9, p. 72 , hence it follows from our reduced system of relations as well. Moreover, all of the ht␣Ž. commute; they generate a Cartan subgroup H of G. Furthermore, the relations y1 2Ž␣ ,  .rŽ  ,  . hŽ. xth␣ Ž. Ž. sx ␣ Ž t . Ž.7 also hold in the Chevalley groupw Ste67, Lemma 20Ž. c , p. 29x , for any ␣,g⌽, not necessarily distinctŽŽ. here ␣,  denotes the usual inner product of roots, so that 2Ž.Ž.␣,  r ,  is always an integer of absolute value F 3.. We now make use of these relations to further shorten our presentation Ž.at the same time adapting it to allow the center to be easily eliminated . Ž. We use <<⌿ additional generators denoted h␣ for ␣ g ⌿ . We add the relations Ž.H0 1 y1 y ␣⌿ h␣sy␣Ž.yy␣␣␣ Žy .y Ž.Ž.y Ž.1yy␣␣ Žy1 .y Ž.1 Žg .; note that the right-hand side is just h␣Ž. , expanded according to Ž. 4 and Ž.5 , and it must be further expanded according to definition Ž A0 .Ž but b12s 1 and b s so most of the above terms do not need any further expansion. ; after that expansion each of these <<⌿ relations has length OmŽ.log p . We also add the relations Ž.H1 h␣,hs1 Ž.␣,g⌿; Ž.H2 y12Ž␣,.rŽ,. hy␣Ž. b h sy ␣Ž. b Ž␣,g⌿,1FFm .. 92 BABAI ET AL. Here we expand the right-hand side ofŽ. H2 using Ž. A0 if necessary. d Ž. However, bs bqd for any and d such that 1 F q d F m;inH2 this occurs with d s 2Ž.Ž.␣,  r ,  , hence < yb␣Ž.␣,ybŽ.s1 Ž.␣g⌿,gÄ41,2 , 1 F F m ; Ž.B0Ј ij yb␣Ž.,ybŽ.sŁyCbbi␣qjŽ.i,j,␣, i,j)0 Ž.Ž.␣,gF,,gÄ41,2 , together with the relations Ž.Ž.H0 , H1 , and Ž.H2 abo¨e. Proof. The relationsŽ.Ž.Ž. A1Ј ,A2Ј, and B0Ј are subsets of the relations Ž.Ž.A1 , A2 , and Ž. B0 from Section 4.2, obtained by simply restricting the values of Žas well as in Ž B0Ј ... The additional relations Ž.Ž. H0 ᎐ H2 hold in the Chevalley group G Ž.as noted earlier in this section , thus to prove that the above relations define G it sufficesŽ. by Theorem 4.2 to show that the remaining relationsŽ.Ž. A1 , A2 , and Ž. B0 follow from the ones above. ␣  Ž. Ž.y1 Ž2 .Ž. In the case s , H2 says that hy␣␣ bh ␣sy ␣b syb ␣q2 Ž.Ž.Ž.assuming that q 2 F m . Thus, the relations A1Ј together with H2 imply all of the relationsŽ. A1 , and similarly the relations Ž A2Ј .imply Ž. A2 . Finally, the relationsŽ. B0Ј imply all relations Ž. B0 , because all other pairs Ž.,can be obtained from the four pairs inÄ4Ä4 1, 2 = 1, 2 , using conjuga- c1 c2 tion by h h␣ , where the integers c12and c are chosen as in the following lemma. LEMMA 5.2. Let ␣ and  be two roots from any root system, with ␣/", and let Ž., be an arbitrary pair of integers. Then there are integers c12, c such that Ž.c1212и2Ž.Ž.␣,r,q2c,2c qc и2 Ž.Ž.,␣ r ␣,␣ gÄ4Ä4y1, y 2 = y 1, y 2. SHORT PRESENTATIONS 93 Proof. Let ²:␣,  denote 2 Ž.Ž.␣,  r ,  and similarly for ²:, ␣ , Ž.Ž²: ²:. and write ¨ c12, c s c 1␣,  q 2c 2,2c 1qc 2,␣ . Then we are ÄŽ. 4 looking at the lattice ⌳ s ¨ c12, cc< 12,cgޚ. Because of the way they arise from a root system, the numbers ²:²:␣,  and , ␣ are either both 0, or else one of them is "1 and the other is 1, 2, or 3 times that oneŽ see wxHum72, Table 1, Sect. 9.4, p. 45. . If they are both 0 the conclusion is clear. Otherwise, replacing  by y if needed, we may assume that ²:␣,s1. If ²:, ␣ s 1, then ¨ Ž.Ž.Ž.Ž.Ž.2, y1 s 0, 3 , ¨ y1, 2 s 3, 0 , and ¨ 1, y1 s Ž.y1, 1 . In this case ⌳ s ÄŽ.x, yx 2 OnŽ иlog p q nm q n и m log p q n и m log p 22 2 qnqnmlog p.Žs Onmlog p .s OŽ.log< 5.2. Killing the Center Finally, we now add one or two more relationsŽŽ of length O log< htŽ.иии ht Ž. Ž.8 ␣1 1 ␣n n 94 BABAI ET AL. q ŽŽ..recall 5 , where ␣1,...,␣n are the fundamental roots in ⌽ and the Ž. elements ti g GF q * are chosen as discussed below. Since is a primitive k rootŽŽ. so every element of GF q * has the form with 0 F k - q y 1. Ž k .Ž.k Ž. Ž. and h␣ s h␣ by 6 , we can express any element 8 in terms of ŽŽ.. Ž . our generators h␣ recall H0 with a relation of length Onlog q s OnmŽ.Žlog p F O log< So it only remains to consider which values of ti putŽ. 8 into the center Z. These are easily found by matrix computations in specific cases; in generalŽ seewx Car72, Sect. 12.1.Ž. , Z consists of exactly those elements 8 for which n 2Ž ␣ ij, ␣ .rŽ ␣ ii, ␣ . Łti s1 for j s 1,...,n.9Ž. is1 Ž. Ž. 2y1 EXAMPLE 5.3. In the case Aqn the equation 9 are tt12 s1, y12y1 Ž.y12 i tttiy1 iiq1s12FiFny1 , and ttny1nis1; or, equivalently, t s t1 Ž.nq1 2FiFnand t1 s 1. From this, given , it is easy to find the Ž. appropriate t1 g GF q that produces a generator of Z. The other groups LnŽ.qcan be handled in a similar manner. Since is a primitive root of GFŽ.Ž. q , 9 amounts to a system of linear equations. Namely, if we let zi denote the integer for which zi tiis and 0 F z F q y 2, thenŽ. 9 is replaced by the following system of n congruences: n 2Ž.␣ij, ␣ Ýzi'0Ž.Ž.Ž. mod q y 1 j s 1,...,n .10 Ž.␣,␣ is1 ii Ž. Ž This system can be solved mod q y 1 for the zi this is especially easy Ž.Ž. . since most of the Cartan integers 2 ␣ij, ␣ r ␣ ii, ␣ are 0 , and the solu- tions give theŽ. at most n q 1 elements of the form Ž. 8 in Z. Using these Ž.or enough to generate Z together with the set of relations in Theorem 5.1, we obtain a presentation of GrZ. This completes the proof of THEOREM 5.4. Conjecture 3 holdsŽ and consequently Conjectures 1 and 2 hold., with C s 1, for simple Che¨alley groups of rank G 2. SHORT PRESENTATIONS 95 6. TWISTED GROUPS OF RANK G 2 The twisted groups have presentations similar to those discussed above for the untwisted groups. Details of Steinberg’s presentations for central extensions of the twisted simple groups can be found inwx Gri73 . These presentations can be reduced in length, and supplemented to eliminate the center, in essentially the same way as described in the previous sections. However, a few additional complications arise, especially for the odd- Ž.2Ž. dimensional unitary groups PSU 2k q 1, q s Aq2 k and the Ree groups 2 2 Fq4Ž., where we only obtain a presentation of length O Žlog < THEOREM 6.1. Conjecture 3 holds for the twisted simple groups of rank G 2 Ž. 2 Ž. 2. It holds with C s 1, except perhaps for the groups A2 k q and F4 q , where it holds with C s 2. This is proved in the following subsections. 6.1. Easier Twisted Groups 2 First, we consider types AqnŽ.with n odd Ž the case of even n will be . 2 Ž.Ž . 2 Ž. discussed in Section 6.3 , Dqn for any n G 4 , and Eq6 . In these cases ⌽ Ž. the ‘‘twisted’’ root system has type Ckq1 where n s 2k q 1,Bny1, and F4 , respectively. In each of these cases the long roots of ⌽ correspond to root subgroups of order q, while the short roots correspond to root subgroups of order q 2 Žand all the root subgroups are elementary abelian as in the untwisted case. . Thus, in these cases we start with generators Ž. Ž . Ž 2. xt␣ , where ␣ g ⌽, and t g GF q if ␣ is long while t g GF q if ␣ is short. The Steinberg relations are then as follows: Ž.A xt␣ Žqu .sxtxu␣␣ Ž. Ž. 2 Ž.␣g⌽,t,ugGFŽ. q or GFŽ. q for ␣ long resp. short ; Ž.B xt␣Ž.,xu Ž . ¡1 for ␣ q  f ⌽, ⑀ ␣  ␣  x␣q␣Ž.tu for , , q all short or all long, s~⑀ ␣  ␣  x␣q␣Ž.Ž.tu q tu for , short, q long, x ⑀tu x tuu for ␣ , ␣ 2 long,  , ␣  short. ¢␣q␣␣Ž.q2␣Ž. q q 2 Here u ¬ u denotes the involutory field automorphism of GFŽ. q , and the coefficients ⑀␣␣and are "1 and depend only on ␣,  Ž.and ⌽ . 96 BABAI ET AL. As in the untwisted case, we only need a subset of the above relations Žbywx Cur65 ; see Remark 3.2. , where we consider only roots ␣ g ⌿ and, for Ž.B , only pairs of roots Ž␣,  .g F, where the subsets ⌿ : ⌽ and F : ⌿ = ⌿ are defined as in Section 4. We then shorten this presentation essentially as in Sections 4.2 and 5. 2 This time we use a basis Ä4b12,...,b mof GFŽ q .Ž.Žover GF p where m. Ä4 Ž. qsp , chosen so that its first half b1,...,bm is a basis of GF q over 2 q 1 GFŽ. p . Specifically, let be a primitive root of GF Ž q .,so q s is a Ž. Ž .y1 primitive root of GF q ; then, for 1 F F m, let b s and bmq s b . As before we define Ž.A0 k yt␣Ž.[ŁÝyb␣ Ž .whenever t s kb Ž0Fk -p ., 2 where t g GFŽ. q and 1 F F m if ␣ is long, but t g GF Ž q .and 1 F F 2m if ␣ is short. Ž.Ž We define a group G0 , generated by the symbols yb␣ ␣g⌿, and 1FFmfor ␣ long, 1 F F 2m for ␣ short. , with additional genera- Ž. tors h␣ ␣ g ⌿ and the following relations: Ž. Ž.p Ž A1 yb␣ s1␣g⌿,s1, 2, and also s m q 1, m q 2if ␣ is short. ; Ž. Ž. Ž. ŽŽ. A2 wyb␣ ␣,ybxs1␣g⌿,as in A1 , 1 F F m or 1 F F2mfor ␣ long resp. short. ; Ž.B0 The relations Ž. B above for pairs Ž␣,  .g F, except with x replaced by y throughout and t, u taking only the values b , b , with ,gÄ41, 2 or Ä 1, 2, m q 1, m q 2 4 as appropriateŽ as before, the right- hand sides ofŽ. B0 must be expanded, using Ž. A0 , into expressions involving the generators yb␣Ž.. ; Ž. Ž.Ž . Ž. H0 h␣ s h␣ ␣ g ⌿ , where the right-hand side of H0 is expanded according to the definitionsŽ.Ž. 5 , 4 , and Ž A0 . as before; Ž. Ž. H1 wxh␣ , h s 1 ␣,  g ⌿ ; Ž . Ž .y12 ŽŽ .Ž␣,.rŽ,. . Ž H2 hy ␣ bh sy ␣ b ␣,  g ⌿;1FFmor 1FF2mfor ␣ long resp. short. As before we expand the right-hand side ofŽ. H2 using Ž. A0 if necessary, Ž. d Ž but b s bqd in most cases whenever and q d are either both between 1 and m or both between m q 1 and 2m., thus, as before, expansion is required only for a bounded number of values of . To see that the above relations hold in G, onlyŽ. H2 requires some additional remarks. Each twisted group is defined as a subgroup of the Ž 2 Ž. Ž2. corresponding untwisted Chevalley group e.g., Aqnn;Aq, and simi- larly in the other cases. . Furthermore, each ‘‘root subgroup’’ of the twisted SHORT PRESENTATIONS 97 group corresponds to parts of one or more root subgroups in the untwisted groupŽ cf.wx Car72, Sect. 13.6 ; for ␣ short, our xt␣Ž.here corresponds to Ž. Ž. Ž. Ž. Ž. xtSrrsxtxt in Carter’s notationw Car72, 13.6.4 iix , and our hu␣ corresponds to huhurrŽ. Ž.wxCar72, 13.7.2.Ž. . Thus the relations H2 can be verified in G by straightforward calculations using the corresponding relations for the untwisted group. However, we can also see that they must have the simple form given above by reasoning as follows: The conjugate y1 huxthu Ž.␣ Ž. Ž. has the form xfut ␣ŽŽ..for some function f, because of the way this occurs inside a larger untwisted group. InŽ. H2 we only need 2 to know fuŽ.for u g GF Ž.Ž q though u g GF Ž q ..may occur in general , and this case has the same formula as in the untwisted casesŽ i.e., 2Ž␣,.Ž,. fuŽ.su r for u g GFŽ. q ; see Ž. 7 in Section 5.1 . . This holds because this formula is determined inside another untwisted Chevalley group, namely the subgroup of our twisted group G obtained by restricting all scalars to GFŽ. q . That gives an untwisted group with root system ⌽ ŽŽ.2Ž. Ž2.Ž.2Ž. Ž2.. e.g., Cqkq12;Aqkq12;Aqkq1466,orFq;Eq;Eq . This can be seen from the fact that it is the intersection of theŽ. untwisted subgroups of fixed points of a pure graph automorphism and a pure field automorphismŽ whereas the twisted group is only fixed by the composition of the graph and field automorphisms; cf.wx Ste67, p. 171. . Alternatively, observe that restricting all scalars to GFŽ. q in the above Steinberg presentation for the twisted group produces the Steinberg presentation for the untwisted group with root system ⌽. Thus, all of the above relations hold in the twisted group G. Then, just as in the previous sections, it is easy to prove that the group G0 defined by the above presentation is isomorphic to the group presented by the Steinberg relations aboveŽ which is a central extension of the twisted simple group in question. . The center Z can be eliminated as before:w Ste67, proof of Theorem 35Ž. b , p. 192; cf. Theorem 34 and following exercise, p. 187x shows that the center of the twisted group G is contained in the center of the correspond- ing untwisted universal Chevalley group. Therefore the elements of Z can be expressed as in Section 5.2, in terms of generators for the Cartan subgroup of the untwisted groupŽŽ which is defined over GF q 2... But Z lies inside G, so in fact it can be expressed in terms of generators for the Cartan subgroup of G, namelywx Car72, 13.7.2 the elements h␣Ž. for ␣ shortŽŽ.Ž.Ž. as noted above, these ht␣ correspond to hthtrrin Carter’s notation. and h␣ Ž .for ␣ long Ž these ht␣ Ž.correspond to those htr Ž. which are contained in the twisted group if and only if t s t.. Thus, for ␣ long, we may use our generators h␣ fromŽ. H0 above, but for ␣ short we X introduce additional generators h␣ and additional relations Ž.X Ž.Ž . H0Ј h␣ s h␣ ␣g⌿,␣short , 98 BABAI ET AL. with the right-hand side ofŽ H0Ј .expanded according to Ž.Ž. 5 , 4 , and Ž A0 . X as before. Then, using these generators h␣ for ␣ short and h␣ for ␣ long, we may kill the center as in Section 5.2. Finally, observe that this presentation again has length OŽlog< 6.2. Triality Twisted Groups 3 The group Dq42Ž., with ‘‘twisted’’ root system ⌽ of type G , is handled similarly. Since ⌽ now has rank 2, we have ⌿ s ⌽ in this case. This time Ž. Ž . we start with generators xt␣ , where either ␣ is long and t g GF q ,or␣ 3 is short and t g GFŽ q .. We use t ¬ t to denote the field automorphism 3 2 of GFŽ q .Ž.of order 3, and we use the notation Tr t s t q t q t for the 3 trace map Tr: GFŽ q .Ž.ª GF q . Then the Steinberg relations are: Ž.A xt␣ Žqu .sxtxu␣␣ Ž. Ž. 3 Ž.␣g⌽,t,ugGFŽ. q or GFŽ. q for ␣ long resp. short ; Ž.B xt␣Ž.,xu Ž . ¡1 for ␣ q  f ⌽, ⑀ ␣  ␣  x␣q␣Ž.tu for , , q long, ⑀ ␣  ␣  x␣q␣Ž.TrŽ.tu for , short, q long, ⑀22 2 x␣q␣ž/ž/Ž.Ž.tu qtu x2␣q␣Tr tt u ␦ 2 ␣  ␣  s~=x␣q2␣ž/TrŽ.tu u for , , q short, 2␣q,␣q2long, ⑀ 22␦ x␣q␣Ž.tu x2␣q␣ž/ž/tt ux3␣q␣tt t u ␥22 ␣ ␣  ␣  =x3␣q2␣ž/2tt t u for , q ,2 q short, ¢ ,3␣q,3␣q2 long. SHORT PRESENTATIONS 99 Again the coefficients ⑀␣␣␣␣, , ␦ , ␥ are "1 and depend only on ␣ and . We shorten this presentation in a way completely analogous to the cases 2 in Section 6.1. This time let be a primitive root of GFŽ q 3.,so is 2 Ž. Ž .y1 a primitive root of GF q , and use the basis b s , bmqs b , 2Ž. Ž.Ž. and b2 mqs b for 1 F F m . Then use relations A1 through H2 just as above, except replacing 2m by 3m ŽandÄ4 1, 2, m q 1, m q 2by Ä41, 2, m 1, m 2, 2m 1, 2m 2. in the obvious places, and replac- q q2 q q ing by inŽ. H0 and Ž. H2 . 3 In this case the universal central extension of Dq4Ž.is actually the 3 simple group itself, since Dq4Ž.has trivial Schur multiplierw KL90, Theo- X rem 5.1.4, p. 173x . Thus, we may omit the generators h␣ and relations Ž.H0Ј in this case. But even if we include them, counting as before, we find that our presentation has length OŽ.Žlog q s O log< 6.3. Odd-Dimensional Unitary Groups 2 The groups AqnŽ.with n even are somewhat harder Ž these are the odd-dimensional unitary groups PSUŽ..Ž n q 1, q . Let n s 2kkG2 since we are only considering rank G 2 in this section. . For many purposes the root system of this twisted group is best viewed as BCk , namely, the union of a Bkkand C system. However, following Griesswx Gri73 , we will just use aCk system ⌽. An additional complication in this case is that the root subgroups are not all abelian. Those corresponding to short roots of ⌽ are elementary abelian of order q 2, but those corresponding to long roots are nonabelian 3 Ž. of order q . We start with generators xt␣ for ␣ g ⌽ short and t g Ž.22 Ž . Ž. GF q , and xt␣ ,ufor ␣ g ⌽ long with t, u g Gq and u q u s ⑀␣ tt, where ⑀␣ s "1 are constants depending on the root system and u ¬ u again denotes the involutory field automorphism of GFŽ q 2 .. For relations Ž.A we have Ž. Ž . Ž.Ž.Ž . A xt␣ qusxtxu␣␣ as before for ␣ short, but for ␣ long we have xt␣Ž.Ž.,ux␣␣¨,wsxtŽ.q¨,uqwq⑀ ␣t¨, Ž.AЈ xt␣Ž.Ž,u,x␣␣␣¨,w .sxŽ.0, ⑀ Žt¨ y t¨ .. ŽSeew Car72, 13.6.4Ž. ivxwx ; the version in Gri73 occasionally has .⑀ instead of our "⑀␣ .. 100 BABAI ET AL. 2 The relationsŽ. B for this case ŽG of type Aq2 kŽ..are as follows: Ž.B xt␣Ž.,xu Ž . ¡1 for ␣ ,  short, ␣ q  f ⌽, x Ž.⑀ tu for ␣ ,  , ␣  short, s~ ␣q␣ q x 0, ⑀ Ž.tu tu for ␣ ,  short, ␣  long, ¢␣q␣Ž.y q xt␣Ž.,u,x Ž¨,w . 1 1 for ␣ ,  long, 2Ž.␣ q  f ⌽, s 1 x1Ž.Ž.⑀t for ␣ ,  long, Ž.␣  short, ½2␣q␣¨ 2 q xt␣Ž.,u,x Ž.¨ ¡1 for ␣ long,  short, ␣ q  f ⌽, ⑀ ␦ s~x␣q␣Ž.u¨ x ␣q2␣␣Ž.t¨ , u¨¨ ¢for ␣ ,␣ q 2 long,  , ␣ q  short. ŽAgain the coefficients ⑀␣␣␣, , ␦ are "1 and depend only on ␣ and .. We will shorten these relations essentially as in Section 6.1, but some differences arise. The reduction from ⌽ to ⌿ Ž.and F is as before. Let Ž 2 . Ä4 be a primitive root of GF q and let b :1FF2m be a basis defined in terms of as in Section 6.1. Let denote a fixed nonzero element of 2 Ž q 1. 2 GFŽ. q for which sy Žif q is odd, then let s q r , while if q is even let s 1.. Then, for a long root ␣, the root subgroup elements of the form x␣Ž.0, u Ž. Ž. can be written as x␣ 0, t with t g GF q ; these elements form a sub- group isomorphic to the additive group of GFŽ. q . Modulo that subgroup, the root subgroup is isomorphic to the additive group of GFŽ q 2 .. More- over, the commutator relations in the root subgroup are determined by the GFŽ. q -bilinear map t¨ y t¨, and hence are completely determined on a basis of GFŽ q 2 .Ž.over GF p . 2 For each long root ␣, let u␣,1 denote a fixed element of GFŽ q . for Ž. which u␣,1 q u␣,1 s ⑀␣bb11s⑀␣ recall that b1s 1 , and similarly let ⑀ ⑀ u␣,mq1be an element for which u␣,mq1q u␣,mq1s ␣bbmq1 mq1s␣ . Ž. Elements u␣,1 and u␣,mq1 with the desired trace can easily be computed. Then define 2 222 y y u␣,␣sŽ.u ,1 and u␣,mq␣s Ž. u ,mq1 for 2 F F m. SHORT PRESENTATIONS 101 Ž.Ž. Thus, we have u␣,␣q u ,␣s ⑀ bb for all 1 F F 2m ,so xb ␣␣,u, is an element of G ŽŽcf. A0Љ .below . . Now for our short presentation we use the following generators. For Ž.Ž . each short root ␣ g ⌿ we use 2m generators yb␣ 1FF2m. For Ž. each long root ␣ g ⌿ we use 3m generators, namely y␣ 0, b for Ž. 1FFmand yb␣ ␣,u,for 1 F F 2m. In addition, our presentation X will use generators h␣ and h␣ for each ␣ g ⌿. For short roots ␣ g ⌿ we defineŽ. as before k Ž.A0 yt␣ Ž.[Łyb␣ Ž. 2 whenever t s ÝkbgGFŽ. q Ž.0 F k - p,1FF2m ; whereas for long roots ␣ g ⌿ we define k Ž.A0Ј y␣ Ž.0, u [ Ły␣ Ž0, b . y1 whenever u s ÝkbgGFŽ.Ž q 0 F k - p,1FFm .; k Ž.A0Љ yt␣ Ž.,u[y␣␣␣ Ž.0,¨ Łyb Ž,u, . Ž.Ž2 . whenever t s Ý kbgGF q 0 F k - p;1FF2m and u q u s Ž.2 ⑀␣ tt, where b␣and u ,are as defined above and ¨ g GF q is the k appropriate element to make this relation hold in G ŽŽ.i.e., Łxb␣,u ␣, Ž. Ž2.Ž. sxt␣ ,wfor some w g GF q which has, from AЈ , a known expression in terms of the b , k␣, u ,␣, and ⑀ ; then let ¨ s u y w so that Ž.Ž. Ž. . Ž. x␣ 0, ¨ xt␣␣,wsxt,uas desired ; the factor y ␣0, ¨ is itself expanded according toŽ. A0Ј . For each short root ␣ g ⌿ we use the relations: p Ž.A1 yb␣ Ž. s1 Žs1,2, m q 1, m q 2; . Ž.A2 yb␣Ž.␣,ybŽ.s1 Žs1,2, m q 1, m q 2, 1 F F 2m .. On the other hand, for each long root ␣ g ⌿ we use the following relations: p Ž.A3 y␣ Ž0, b .s 1 Ž s 1,2,3,4 . ; Ž.A4 y␣Ž.0, b␣ , y Ž.0, b s 1 Ž s 1,2,3,4, 1 F F m .; 102 BABAI ET AL. p 1 for p ) 2, Ž.A5 yb␣ Ž␣,u, .s ½y␣Ž.0, ⑀␣bb for p s 2 Ž.s1,2, m q 1, m q 2, Ž. Ž. where, for p s 2, y␣ 0, ⑀␣bb is expanded according to A0Ј ; Ž.A6 yb␣ Ž␣,u,␣␣ .,ybŽ.,u,␣␣syž/0, ⑀ ž/bb ybb Ž.s1,2, m q 1, m q 2, 1 F F 2m , where again the right-hand side is expanded according toŽ. A0Ј ; Ž.A7 yb␣Ž.␣,u,␣,yŽ.0, b s 1 Ž s 1,2, m q 1, m q 2, 1 F F m .. We use the following relations in place ofŽ. B : Ž.B0 ¡1 for ␣ ,  short, ␣ q  f ⌽, ⑀ ␣  ␣  y␣q␣Ž.bb for , , q short, yb␣Ž.,ybŽ.s~ y0, ⑀ bb bb ␣q␣ž/ž/y ¢ for ␣ ,  short, ␣ q  long Ž.,gÄ41,2, m q 1, m q 2, y␣Ž.0, b , y Ž.0, b s 1 for ␣ ,  long Ž. , g Ä41,2,3,4 , y␣Ž.0, b , ybŽ.,u,s1 for ␣ ,  long Ž. s 1,2,3,4, s 1,2, m q 1, m q 2, yb␣Ž.␣,u,,yŽ.0, b s 1 for ␣ ,  long Ž. s 1,2, m q 1, m q 2, s 1,2,3,4 , yb␣Ž.␣,u,,ybŽ.,u 1 1 for ␣ ,  long, 2Ž.␣ q  f ⌽, s 1 y1Ž.Ž.⑀bb for ␣ ,  long, Ž.␣  short ½2␣q␣ 2 q Ž.,gÄ41,2, m q 1, m q 2, yb␣Ž.␣,u,,ybŽ. ¡1 for ␣ long,  short, ␣ q  f ⌽, ⑀ ␦ s~y␣q␣␣Ž.uby, ␣q2␣␣␣ž/bb, ubb, ¢ for ␣ , ␣ q 2 long,  , ␣ q  short Ž.,gÄ41,2, m q 1, m q 2, SHORT PRESENTATIONS 103 ¡1 for ␣ long,  short, ␣ q  f ⌽, ⑀ ␦ y␣Ž.0, b , yb Ž.s~y␣q␣␣ž/b byq2␣ž/0, b bb ¢for ␣ , ␣ q 2 long,  , ␣ q  short Ž. s 1,2,3,4, s 1,2, m q 1, m q 2 ŽŽ.Ž.Ž.as before, the right-hand sides of B0 are expanded using A0 , A0Ј , and Ž.A0Љ where necessary . . To define the elements ht␣Ž., for short roots ␣ we can use Ž. 4 and Ž. 5 Ž.see Section 5.1 as before, while for long roots ␣ we use the variation y1 y1 y1 wt␣Ž.,usyt␣ Ž.,uyy␣␣Ž.ytu , u y Ž tu u, u . Ž.11 and y1 ht␣Ž.sw␣␣ Ž¨,t .w Ž0, .,12 Ž . Ž.2 Ž. where t, u g GF q with u q u s ⑀␣ tt as usual, and, in 12 , ¨ can be Ž.2 Ž . taken as any element of GF q for which ⑀␣¨¨ s t y t wSte81, 5.3, 5.8, and 5.6x ; in particular, ¨ s 0if tgGFŽ. q . Given this, the remaining relations we need are the following: Ž.H0 h␣s h␣ Ž. Ž ␣ g ⌿ .; X Ž.H0Ј h␣s h␣ Ž.Ž␣g⌿ . Ž.where the right-hand sides are expanded as discussed above ; Ž.H1 h␣, hs 1 Ž␣ ,  g ⌿ .; 2Ž.Ž.␣,, y1 r Ž.H2 hy␣ Ž. b h sy ␣ž/ Ž. b Ž.␣,g⌿,␣short, 1 F F 2m , 4Ž.Ž.␣,, y1 r hy␣Ž.0, b h s y ␣ž/0, Ž. b Ž.␣,g⌿,␣long, 1 F F m , 2Ž.Ž.␣,, 4 Ž.Ž.␣,, y1 r r hy␣Ž. b,uh ␣,sy ␣ž/ Ž. b , Ž. u ␣, Ž.␣,g⌿,␣long, 1 F F 2m , where the right-hand sides are expanded according toŽ.Ž. A0 , A0Ј , and Ž.A0Љ when necessary, but as before Ž because of our choices of b and u␣, . expansion is needed only for a bounded number of values of . 104 BABAI ET AL. The fact that the relationsŽ. H2 hold in the twisted group G can be seen essentially as in Section 6.1. For ␣ long, our xt␣Ž.,ucorresponds to Ž . Ž. Ž. Ž . Ž. xtS,usxtxtx rrrqr uin Carter’s notationw Car72, 13.6.4 ivx , so its conjugate by hw Ž.has the form xfwt␣ ŽŽ.,fЈ Ž..wu for some functions f and fЈ. As before, forŽ. H2 we only need these functions for w g GF Ž. q , where they are determined inside a smaller untwisted group and fwŽ.has 2 the same form as before. Finally, note that fЈŽ.w s fw Ž.in this situation ŽŽ.Ž.Ž.. for w, fw,fЈwgGF q , because u q u s ⑀␣ tt and the same relation must hold between fЈŽ.wuand fwt Ž.. As before, the above relations imply the rest of the Steinberg relations for the twisted group G ŽŽ.again using Lemma 5.2 to show that H2 produces all other relevant values of , from the ones indicated in the relationsŽ. A1 through Ž. B0 , and then proving essentially as in Section 4.2 that, in each rank 2 subgroup, the Sylow p-subgroup has the appropriate order, hence all relationsŽ. B follow . . Finally, observe that most of the above relations have length OŽlog< 2 6.4. The Ree Groups F4Ž. q 2 Ž. Finally, the remaining twisted groups of rank G 2 are the groups Fq4 of characteristic 2. Here the ‘‘twisted root system’’ is not one of the usual root systems, but has 16 ‘‘roots,’’ corresponding to the 16 vertices 1, . . . , 16 of a regular 16-gon. These alternate long and short around the 16-gon, 2 e 1 with odd indices short and even ones long. Also q s 2 q for some Ž 2 Ž. integer e we may assume e G 1 here, treating Tits’ group F4 2 Ј as 2 sporadic. . Let be the field automorphism such that 2 s 1Ž i.e., 2 e tst for t g GFŽ.. q . The presentation from Griesswx Gri73 for this group is the following: the Ž. Ž . generators are xt␣ for each of the 16 roots ␣, where t g GF q . The relationsŽ. A are of two sorts, depending on whether ␣ is short Ž odd numbered.Ž or long even numbered . . If ␣ is long, then the corresponding root subgroup is elementary abelian of order q, i.e., isomorphic to the additive group of GFŽ. q , so its relations are as before. If ␣ is short, then ²Ž. Ž.: the corresponding root subgroup xt␣ < tgGF q is isomorphic to a Ž 2 .Ž.2Ž. Sylow 2-subgroup of order q of a Suzuki group Sz q s Bq2 . Its 2 center consists of the elements xt␣Ž., and is isomorphic to the additive SHORT PRESENTATIONS 105 group of GFŽ. q , as is the central quotient group. Only a basis of the field GFŽ. q over GF Ž.2 is needed for a presentation of this 2-group. The relationsŽ. A for short ␣ are 2 2q12 Ž.Axtxu␣ Ž.␣␣ Ž.sxt Žqux . ␣ Ž.¨ , where ¨ s tu , 2 2q122 xt␣Ž.,xu␣␣ Ž .sx Ž¨ ., where ¨ s tu q tu, 2 xt␣Ž.,xu␣ Ž . s1, 22 2 xtxu␣Ž.␣␣ Ž .sxt Žqu .. 21 ŽNote that an element ¨ for which ¨ qhas a desired value can easily be computed using a primitive root of GFŽ.. q . The relations of typeŽ. B are the following, together with those obtained from them by transforming the subscripts using the Weyl group of order 16 generated by ␣ ª ␣ q 2 and ␣ ª y␣ Ž.modulo 16 . Ž.B xt12 Ž.,xu Ž.s1, xt13Ž.,xu Ž .sxtu 2 Ž ., 2 2q1 xt14Ž.,xu Ž .sx 3 Ž¨ ., where ¨ s tu, xt15Ž.,xu Ž . s1, 222 xt16Ž.,xu Ž .sx 3 Ž¨ .xtuxw 4 Ž . 5 Ž ., 2 1 2 1 2 1 2 where ¨ q s t q u and w q s tu , 3 q12 2 2q1 xt17Ž.,xu Ž .sxt 2 Ž uxtuxtuxtu . 3 Ž . 5 Ž . 6 Ž ., 2 2q22q14q22q1 xt18Ž.,xu Ž .sxt 2 Ž uxt . 3 Ž uxt . 4 Ž u . 3 2q12 2q22q1 =xt56Ž.Ž.Ž. u xt u xtu 7, xt24Ž.,xu Ž . s1, xt26Ž.,xu Ž . s1, 2 2 xt28Ž.,xu Ž .sxtuxtu 4 Ž . 6 Ž .. Using a suitable basis for GFŽ. q over GF Ž.2 , this can be reduced to a presentation of length OŽlog2 < 7. GROUPS OF RANK 1 Ž. Ž . The rank 1 universal Chevalley groups are the groups Aq1 sSL 2, q ; their central quotients PSLŽ.2, q are the corresponding simple Ž if q ) 3 . Chevalley groups. Presentations for the groups G s PSLŽ.2, q were found m by Toddwx Tod36 using m q 2 generatorsŽ recall that q s p .; the length 2 2 of his presentation is ⌰Žm log p q m .ŽF O log < GeneratorsŽ. Todd’s notation : U, R, SiiŽ.s0,...,my1. m Žpy1.r23 22p Relations: R sU sŽ.UR s ŽUS0 .s Sis 1 Ž.is0,...,my1, SSijsSS ji Ž. i,js0,...,my1, RSiis SRq2 Ž. is0,...,my3, 3 Ž.SRU1 s1, a a a 0 1 иии my 1 RSmy201s S S Smy1R, b b b 0 1 иии my 1 RSmy101s S S Smy1R. When p s 2, Todd’s presentation is of similar nature. A shorter presenta- tion for this case was found by Sinkovwx Sin39Ž who also mentioned that something similar could be done to shorten Todd’s presentation for odd mm p.Ž.Sinko¨ ’s presentation for G s SL 2, 2 .Žs PSL 2, 2. is the following: Generators: x, y, z 2m 123 22i2 Relations: x ys y s z s Ž.xz s Ž.yz s x , y s 1 Ž.is1,...,my1, ma a a xsyxy my 1 x иии y 1 xy 0 , ŽŽ.. where the aiiare defined as above a g GF 2 . The length of this 2 presentation is ⌰Ž.Žm log m s ⌰ log< p 243 Žp1.22 PSLŽ.2, p s¦;x, y : x s y s Ž.xy s Ž.xyxqr y s1. Remark 7.3. More recently Campbell, Robertson, and Williams wxCRW90 have found shorter presentations than Todd’s and Sinkov’s for PSLŽ2, p m.. Their presentations can be written so as to have length ⌰Žlog< 8. SHORT PRESENTATIONS FOR ALL FINITE GROUPS In this section, we prove Theorem 1.4. It will be important to reach rapidly any element of a group from any set of generators. This is always possible by the next lemma. Let G be a group, and A and T two subsets of G.A straight line Ž. program, computing A from T, is a sequence S s w1,...,ws of elements of G such that A ; S Žwe use the same symbol S to denote the set Ä4w1,...,wsi., and each member w of S is either a member of T,ora y1 product wwjkfor some j, k - i,or w j for some j - i. The length of S is s.If tof the elements in the sequence S belong to T, then the reduced length of S is s y t Ž.we do not count accesses to the generators . The cost of A relative to T, costŽ AT< ., is the shortest reduced length of straight line programs computing A from T.Ž For a single element A s Ä4g we abbreviate costŽÄ4gT<.Žas cost gT<..IfTdoes not generate A, the cost is ϱ.. We note that for any sets A, B, C : G we have costŽ.AC<< LEMMA 8.1Ž. Reachability Lemma . Let G be a finite group, T a set of 2 generators, and g g G. Then costŽ gT< .Ž- 1qlog< LEMMA 8.2. Let G be a finite group and T a set of generators. Then there exists a set A ; G such that: Ž.i cost ŽAT< .-log2< THEOREM 8.3. If each composition factor Hi of the finite group G has a Ž C . presentation of length O log < Proof. We shall use the terms ‘‘short’’ and ‘‘efficient’’ in a formal sense. A set A of generators is efficient if it satisfies the boundŽ. 14 . A straight line program which computes an efficient set of generators is short if it satisfies the boundŽ. 13 . A straight line program computing an element from an efficient set of generators is short if it satisfies the boundŽ. 14 . Ž Let G s G01dG d иии dGms 1 be a composition series in G note that <<.Ž. mFlog G and let Hiidenote the simple factor group G y1rGi. Let GenŽ.Hiiand Rel Ž.H denote the sets of generators and relations, respec- C tively, of a presentation of Hiihaving length OŽlog < 1 R1. If some t g T arose as t s ¨w or t s wy in the course of one of the straight line programs Si, DwŽ.,P Ž.,orQw Ž,z .referred to above, we include the relation xtŽ.sx Ž¨ .xw Ž .or xtxw Ž. Ž .s1, respectively. Ž. Ž . R2. For each g Rel Hi , lifted to w1,... sz as above, we ŽŽ . . Ž . include the relation xw1,... sxz . R3. For w g L0, i and z g L0, j, i - j, we include the relation 1 1 xwŽ y zw.Ž.sxwy xzxwŽ.Ž .. 110 BABAI ET AL. Let Gˆ be the group defined by these relations. We clearly have a homomorphism : Gˆ ª G onto G mapping xtŽ.to t g T. We claim is one-to-one. It suffices to show that < CSŽ.FÝsetup ŽGi .F m и setup Ž max . , i CDŽ.FÝÝGen Ž.Hiireach Ž.G F reach Ž max . Gen Ž.H i, ii CPŽ.FÝÝRel Ž.Hiireach Ž.G F reach Ž max . Rel Ž.H i, ii 2 CQŽ.FÝlog< Ž. Ž. Ž. where setup max s maxii setup G , and reach max is defined analo- gously. The contribution of R2 is the sum over all i of the total lengths of the relations in RelŽ.Hi . The contribution of R3 is OŽlog2 < Now observe that< GenŽ.Hii< Omlog2 < In particular, we have COROLLARY 8.4. If the finite group G has no composition factor which is a rank 1 twisted group of Lie type, then G admits a presentation of length OŽlog3 < COROLLARY 8.5. E¨ery sol¨able group G has a presentation of length OŽlog3 < PROPOSITION 8.6. Let minpresŽ.G denote the length of the shortest presentation of the group G. For a positi¨e integer N, let minpresŽ.N s n maxÄ minpresŽ.G : < 3 ApŽ.n )pŽ2r27y⑀ n.n ,15 Ž. where ⑀ depends on n but not on p and lim ⑀ 0. On the other nnªϱns hand, the number of groups admitting presentations of length k Žincluding k parentheses and other special characters. is - k . Thus, for k s 4 и minpresŽ p n. we have k k ) ApŽ.n .16 Ž. A comparison ofŽ. 15 and Ž. 16 proves Proposition 8.6. The possibility of improving the reachability lemma remains open. Problem 8.7. Is it possible to improve the OŽlog2 < ACKNOWLEDGMENTS We are indebted to J. H. Conway and A. 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