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On the Diameter of Finite Groups

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On the Diameter of Finite Groups On the diameter of finite groups L. Babai G. Hetyei University of Chicago Mathematical Institute of the and Hungarian Academy of Sciences Eotvos University, Budapest, Hungary and M. I. T. W.M. Kantor A. Lubotzky University of Oregon, Eugene, OR Hebrew University, Jerusalem, Israel A. Seress Ohio State University, Columbus, OH Abstract 1 Introduction The diameter of finite groups has been investigated in connection with efficient communication networks (cube-connected cycles, etc. [Sto], [PV] and others) and generalizations of Rubik’s puzzles (cube, rings) ([DF], [McK]). Determining the exact diameter with The diameter of a group G with respect to a set S of respect to a given set of generators is NP-hard even generators is the maximum over g G of the length E in the case of elementary abelian 2-groups (every ele- of the shortest word in S U S-’ representing g. This ment has order 2) (Even, Goldreich [EG]). The length concept arises in the contexts of efficient communica- of the shortest positive word (no inverses) represent- tion networks and Rubik’s cube type puzzles. “Best” ing an element in terms of given generators of a per- generators (giving minimum diameter while keeping mutation group is known to be PSPACE-complete the number of generators limited) are pertinent, to (Jerrum, [Je]). The difficulty of the problem of deter- networks, “worst” and “average” generators seem a mining the diameter is indicated by the gap between more adequate model for puzzles. We survey a sub- the best known upper and lower bounds for Rubik’s stantial body of recent work by the authors on these cube (see [FMSST]). subjects. Regarding the “best” case, we show that while the structure of the group is essentially irrele- In this paper we report some progress on estimating vant if IS1 is allowed to exceed (logIGI)’+C (c > 0), the best, worst, and average case diameters of various it plays a heavy role when 1.51 = O(1). In particu- classes of finite groups. In connection with the average lar, every nonabelian finite simple group has a set of case problem, we consider the probability that a small 5 7 generators giving logarithmic diameter. This can- set of random elements generates a given group. not happen for groups with an abelian subgroup of The results were obtained by various subsets of the bounded index. - Regarding the worst case, we are authors. The full proofs will appear in a number of concerned primarily with permutation groups of de- separate papers [Ba2], [BH], [BKLZ], [BS2], [Ka], [KL]. gree n and obtain a tight exp((nlnn)’/2(1 + o(1))) upper bound. In the average case, the upper bound Let G be a finite group and S a set of generators improves to exp((lnn)’(l + o(1))). As a first step to- of G. The diameter diam(G, S) is the maximum over ward extending this result to simple groups other than g E G of the minimum word length expressing g as A,, , we establish that almost every pair of elements of a product of elements of S U S-*. Taking the posi- a classical simple group G generates G, a result previ- tion that we wish to minimize the diameter, we de- ously proved by J. Dixon for A,. In the limited spa.ce fine the worst case diameter of G to be diam,,,G = of this article, we try to illuminate some of the basic maxs diam(G, S) (the generators are brought in by an underlying techniques. adversary). 857 CH2925-6/90/0000/0857$01.OO (Q 1990 IEEE We have to be a little careful when defining the best We are unable to find two generators which would prG case: when referring to friendly generators (selected by duce optimal (O(1og [GI)) diameter, but 7 generators ourselves), there should be a limit on how many gen- suffice for this purpose: erators we are entitled to use. We use diamm,,,(G, k) to denote the minimum of diam(G,S) Over all sets of Theorem 2.1 [BKLZ] Every nonabelian finite simple generators. (If do not suffice to generate group G has a set S of at most 7 generators such that G then this quantity is infinite.) the resulting diameter is O(log IGl). Similarly in the average case: we consider bounds on diam(G,S) which hold for almost every choice of In other words, diam,i,(G, 7) = O(1og IGl). This re- the k-set S c G. (This is not to be confused with the sult is algorithmically eficient; given an element of G, average diameter over the same collection of Cayley the corresponding word can be found using O(1og IGl) graphs; we do not invesigate this quantity here.) group operations. (Here we assume that the elements of G are given in their "natural" matrix representa- In this connection it is of interest to find out what tion.) the minimum number of generators is, and to find the tradeoff between the size of a random subset S of a A comment on expanders is in order here. For sim- group and the probability that S generates G. plicity, we use the term for the not necessarily bipar- tite version: an €-expander is a graph in which the We note that in the network context one is most boundary of every set A of at most half of the vertices interested in the best case; in the puzzle context, the has size 2 CIA(.A family of expanders is a family of worst and average cases seem more relevant. Fam- €-expanders for a fixed 6 0. ilies of expanders are relevant for the best case; we > shall comment on this connection after the statement Expanders on n vertices have diameter O(1og n). of Theorem 2.1. The current champion of bounded degree explicit ex- panders arises as a family of Cayley graphs of the simple groups PSL(2, q) (Margulis [Ma], Lubotzky - 2 Statement of the main re- Phillips - Sarnak [LPS]). Some other families of fi- nite simple groups have also been known to give rise sults to families of expanders (notably, PSL(d,q) for fixed dimension d 2 3, see Alon - Milman [AM]). It is The Cayley graph r(G,S) of the group G wit.h re- not known, however, whether or not all finite simple spect to the set S of generators has G for its vertex groups admit bounded degree expander Cayley graphs. set; the edges are the pairs {g,gs} (g E G, s E S). Even the seemingly most accessible cases are open. The diameter of this graph is diam(G,S). Note tlmt diam(G, S) = diam(G, S U S-'). Problem 2.2. Do the alternating groups and the linear groups PSL(d,q) for fixed q admit families of bounded degree expander Cayley graphs? For obvious counting reasons, the diameter is at least logarithmic as a function of the order of the group: We should also point out that the known construc- tions of explicit expanders do not yield the algorithmic diam,i,(G, k) > log IGl/ log(2k). (1) consequence stated after Theorem 2.1. (Observe that in comparison to the size of the Cayley graphs con- This bound is tight, up to a constant factor, for structed, we find short paths in logarithmic number of every group G, assuming k > (loglGl)'+' for some group operations.) positive constant c (Prop. 3.2). On the other hand, the ratio of the two sides of inequality (1) may diverge The number 7 in Theorem 2.1 can probably be im- while IGI +. 00 if k 5 (log IGl)'+"(') (Prop. 3.3). proved. The following result exists in this direction. The greatest interest is in the case of groups gener- ated by a bounded number of elements. It is natural 2'3 [Ka] 'f 2 lo then there IW0 q) Ihe diameter to begin the ''best case" study of these groups wit]>the generators Of = PSL(n, 'Iiat is finite simple groups. These groups turn out to behave '(log IGI>. One Of generators be quite favorably. to be an involution (i.e. an element of order two). By a result essentially due to R. Steinberg, every fi- (This reduces the degree of the Cayley graph, i.e. the nite simple group is generated by two elements ([Ste]). size of the set S U S-' to 3 which is minimum.) The 858 same result holds for the other classes of classical sim- Theorem 2.9 [Ball The probability that a randomly ple (matrix) groups of sufficiently large dimension, as selected pair of permutations from S, generates A, or well as for the alternating groups [Ka]. S, is 1 - 1/n + O(~I-~). We suspect that finite simple groups behave quite While Dixon's proof is completely elementary, unfortu- nicely even in the worst case. More precisely, the fol- nately the proof of this result, like the proofs of nearly lowing conjecture has been made. all results announced here, depends on the classifica- tion of finite simple groups. (Theorems 2.5 and 2.10 Conjecture 2.4 [BSl] For every finite simple group are the only exceptions.) The best error-term obtained G, diammax(G)= (log IGI)O('). previously using generating function techniques only was n-'+'(') (Bovey [Bo]). Unfortunately we are unable to verify this conjecture We are able to prove that a random pair of per- even in the case of the alternating groups A, of order mutations not only generates A, or S,, but that it n!/2.
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