Hans Zassenhaus, "Uber transitive Erweiterungen gewisser Gruppen aus Automorphismen endlicher mehrdimensionaler Geometrien", Math. Ann., Ill (1935) ...
ON THE ORDERS OF PRIMITIVE PERMUTATION GROUPS
CHERYL E. PRAEGER AND JAN SAXL
The problem of bounding the order of a permutation group G in terms of its degree n was one of the central problems of 19th century group theory (see [4]). It is closely related to the 1860 Grand Prix problem of the Paris Academy, but its history goes in fact much further back (see e.g. [3], [1] and [10]). The heart of the problem is of course the case where G is a primitive group. The best result here is due to A. Bochert [2]: if G is primitive and An ^ G then \Sn: G\ ^
—— !. In the
special case where G is not 2-transitive this was greatly improved by Wielandt [13]: in this case \G\ ^ c" for some constant c (independent of n and G - i n fact c = 24 was shown to be sufficient). However, the assumption that G is not 2transitive was essential in Wielandt's approach. It is our aim in this paper to remove this. THEOREM. There is a constant c such that \G\ < c" for any primitive group G of degree n not containing the alternating group An, and for any n. In fact we can take c = 4.
The notation we use is fairly standard (see [12]), but we shall use the symbol GA to denote the setwise and G(A) the pointwise stabilizer in G of a set A. If a and b are (a) integers with b j= 0, then is the least integer m such that m ^ a/b. Also if b is a positive integer and p is a prime, bp will denote the highest integer m for which pm divides b. The proof of the theorem relies on the following two propositions (see [13, Lemma 8.6 and the proof of Theorem 8.5]). PROPOSITION 1. Let G be a primitive group of degree n which is t-transitive but not (t + intransitive, with 1 ^ t < n — 2. Let p be a prime number with p2 ^ n and let Q be a p-subgroup of G all of whose orbits have length 1 or p. If Q has order pf then
PROPOSITION
2. Let G and p be as in Proposition 1. If the Sylow p-subgroups have
, ft —11 n order pe then e ^ < > + -=• + =+ J
[ P )
PP
n
. P(P-1)
Received 19th October, 1979. This paper was written while the second author was visiting the University of Western Australia. We wish to thank the University for making the visit possible. [BULL. LONDON MATH. S O C , 12 (1980), 303-307]
304
CHERYL E. PRAEGER AND JAN SAXL
Proof of Proposition 1 is by induction on n. Notice that the induction does get started, since if n = p2 and \Q\ > p then it follows from [7] that G is 2-transitive and
1(21 = p2Let X be the stabilizer in G of the points a x ,..., /p}+b/p\
It is therefore sufficient to show that \
\^
I PJ
=—. This certainly holds if
P
s < p + 1, so suppose that s ^ p + 2 ^ 5 . If a = 1 and \R\ > p, we get a contradiction from [8] (since the point wise stabilizer of Bl is then nontrivial). If d = 1 then A^,1 is 5-transitive and hence Y^\ is 4-transitive, contradicting [5, Theorem D]. Hence we may assume that ad ^ 4, and so p ^ 5 and b ^ 25. Now s ^ 3 logfc by [11], so to show that < >^ it suffices to show that b p I P ) 2 log b ^ - . Since b ^ p 2 , this is certainly true for all b $s 25. p 3 Proposition 1 is now proved. Proof of Proposition 2. Let P be a Sylow p-subgroup of G. In [13, Lemma 8.6] it is proved that P contains a subgroup Q of index at most pniv(p~l) all of whose orbits have length 1 or p. The assertion now follows from Proposition 1. Proof of the theorem. Let G be a primitive group of degree n other than An or Sn. Write \G\ = a(G). b{G), where a(G) is divisible only by primes less than or equal to yfn, and b(G) is divisible only by primes greater than y/n. We shall bound a(G) and b(G) separately by ant and bn respectively, where G is f-transitive but not transitive. Consider b(G) first. Using [7], [9], and [12, 13.10] we have
n p)(h n p)Cn n n-2^p^n
/ \ q = 2 n-l/q sj p « n/q
/ \q = 8 n/q-5/2 H
306
CHERYL E. PRAEGER AND JAN SAXL
7! r = —exp(( x /n-6.5)logn + 2 = bn,
say.
(In the products p denotes a prime number and q denotes an integer. The last inequality comes from an application of the Stirling formula.) Assume now that G is simply primitive, that is, that t = 1. By Proposition 2,
l08P+ 1
2p-l ,
1999 ? logx
J
1000
< 1 . 2544, and so anA < (3 . 5057)". (Notice that if n is small then (log anl)/n is considerably smaller than 1.2544. We use this fact later.) One checks easily that bn < ( 1 . 7114)" for all n, so that if G is simply primitive then certainly \G\ < 6". Assume now that t ^ 2. Then,
\G\ ^ n{n-l)...{n~
a
By [13] we know that t < 3 log n, and by Proposition 2 we have
n,2
El
Since f ] p < 4 ^ , (cf. [13, p. 41]), it follows that \G\ < n 3 l o g n . 4 ^ . a n > 1 . 6 n . It follows immediately that, for example, \G\ < 22" for any n, and that "asymptotically" we have \G\ < (3 . 5057 + e)" for any positive e. Finally, to show that \G\ < 4" for all n, we first check that this is true for n ^ 12145 using the above estimates for ant and bn, and then, using improved estimates for an2, we check it easily for n ^ 6500. Next, noticing that for reasonably small n and for large t there are trivial extensions of the result [12, 13.10] we can check the bound easily for n ^ 3000. Finally an elementary, but somewhat tedious, computation establishes the bound for the remaining small values of n.
ON THE ORDERS OF PRIMITIVE PERMUTATION GROUPS
307
Remarks 1. An inspection of some families of primitive permutation groups suggests that there should be a bound of the form \G\ ^ c^"108" (cf. [13J, p. 42). 2. Since we have completed our work, Laszlo Babai has improved Wielandt's result: if G is primitive but not 2-transitive then \G\ ^ exp (4^/n Iog2«). 3. Even more recently, Peter J. Cameron has observed that if one is willing to assume the complete classification of finite simple groups then the primitive groups G of degree n for which |G| > clog2" can all be listed. It would be nice to prove similar results without assuming the classification theorem.
References 1. J. Bertrand, "Memoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme", J. Ec. Polyt., 30 (1845), 123-140. 2. Alfred Bochert, "Ueber die Zahl der verschiedenen Werthe, die eine Function gegebener Buchstaben durch Vertauschung derselben erlangen kann", Math. Ann., 33 (1889), 584-590. 3. A. L. Cauchy, "Sur le nombre des valeurs qu'une fonction peut acquerir, lorsqu'on y permute de toutes les manieres possibles les quantites qu'elle renferme", J. Ec. Polyt., 17 (1815), 1-28. 4. Peter M. Neumann, "A 19th Century problem of group theory", in preparation. 5. Michael E. O'Nan, "Normal structure of the one-point stabilizer of a doubly-transitive permutation group, II", Trans. Amer. Math. Soc, 214 (1975), 43-74. 6. Michael E. O'Nan, "Estimation of Sylow subgroups in primitive permutation groups", Math. Zeitschr., 147(1976), 101-111. 7. Cheryl E. Praeger, "Primitive permutation groups containing an element of order p of small degree, p a prime", J. Algebra, 34 (1975), 540-546. 8. Cheryl E. Praeger, "Doubly transitive automorphism groups of block designs", J. Comb. Th., (A), 25 (1978), 258-266. 9. Cheryl E. Praeger, "On elements of prime order in primitive permutation groups", J. Algebra, 60 (1979), 126-157. 10. J. A. Serret, "Memoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme", J. Math. Pures Appl., 15 (1850), 1-44. 11. H. Wielandt, "Abschatzungen fur den Grad einer Permutationsgruppe von vorgeschriebenem Transitivitatsgrad", Schr. math. Sem. Inst. angew. Math. Univ. Berlin, 2 (1934), 151-174. 12. Helmut Wielandt, Finite permutation groups (Academic Press, New York and London, 1964). 13. Helmut Wielandt, Permutation groups through invariant relations and invariant functions, (Lecture Notes, Ohio State University, Columbus, 1969). 14. Hans Zassenhaus, "Uber transitive Erweiterungen gewisser Gruppen aus Automorphismen endlicher mehrdimensionaler Geometrien", Math. Ann., I l l (1935), 748-759.
University of Western Australia, Perth, Australia 6009.
University of Glasgow, Scotland. Present address: New Hall, Cambridge.
