Minimal degree, base size, order: selected topics on primitive permutation groups

In this survey article, we discuss the minimal degree, the base size, and the order of a finite primitive permutation group, along the lines of an article by Martin W. Liebeck.


Orders.
Let G be a primitive permutation group of degree n different from the symmetric group S n and the alternating group A n .How large can the order |G| of G be?This question was raised in the nineteenth century.For interesting historical accounts, see [17,Section 4.10] and [1].Apart from some early results of Jordan, probably the first successful estimate for the order of G was obtained by Bochert [6] (see also [19] or [41]): if G is a primitive permutation group of degree n and different from S n and A n , then (S n : G) ≥ [ 1 2 (n + 1)]!.This bound is good for very small degrees n.
Based on Wielandt's method [42] of bounding the orders of Sylow subgroups, Praeger and Saxl [36] obtained an exponential estimate, 4 n .Using entirely different combinatorial arguments, Babai [1] obtained an e 4 √ n ln 2 n estimate for uniprimitive (primitive but not doubly transitive) groups.(A permutation group G acting on a set Ω is doubly transitive if for any two tuples (α 1 , β 1 ) and (α 2 , β 2 ) from Ω × Ω such that α 1 = β 1 and α 2 = β 2 , there exists a permutation g ∈ G with α 1 g = α 2 and β 1 g = β 2 .Doubly transitive groups are primitive.)For the orders of doubly transitive groups not containing A n , Babai [2] obtained the bound exp(exp(c √ ln n)) for some universal constant c.This was improved by Pyber [37] to an n c•ln 2 n bound by an elementary argument (using some ideas of [2]).Apart from O(ln n) factors in the exponents, the estimates in [1,37] are asymptotically sharp.
To do better, one may want to use the O'Nan-Scott theorem and the classification of finite simple groups.If H is a permutation group acting on a set Γ, then the wreath product H S r acts in a natural way on the set Γ r .This action is a product action.In 1981, Cameron [15] proved the following theorem.Theorem 2.1 (Cameron [15]).There is a (computable) constant c with the property that, if G is a primitive permutation group of degree n, then at least one of the following holds.
(1) G has an elementary abelian regular normal subgroup.
(2) G is a subgroup of Aut(T ) S r , containing T r , where T is either an alternating group acting on k-element subsets, or a classical simple group acting on an orbit of subspaces or (in the case T = PSL d (q) where d is an integer and q is a prime power) pairs of subspaces of complementary dimension, and the wreath product has the product action.
Groups in (1) of Theorem 2.1 have size at most n 1+log n .(The bases of our logarithms will be 2.) It follows from Liebeck's paper [28] that, if G is a group satisfying (2) with T a classical simple group, then |G| < 9 log n.In this context, see also [17,Theorem 4.13] and its proof.
A primitive permutation group is said to be almost simple if it has a unique minimal normal subgroup and that is nonabelian and simple.The class of almost simple primitive permutation groups plays a fundamental role in bounding the orders of primitive groups.The previously mentioned general upper bound 9 log n of Liebeck, with known exceptions, for the order of a primitive permutation group of degree n was obtained from the following theorem.Theorem 2.2 (Liebeck [28]).Let G be an almost simple primitive permutation group of degree n with minimal normal subgroup T. At least one of the following holds.
(1) T = A m acting on k-subsets of {1, . . ., m} or on partitions of {1, . . ., m} into a subsets of size b, (2) T is a classical simple group acting on an orbit of subspaces of the natural module, or (in the case T = PSL d (q) where d is an integer and q is a prime power) pairs of subspaces of complementary dimensions.
The bound |G| < n 9 in part (3) of Theorem 2.2 may be replaced by |G| ≤ n c where c = 6.077948094 is adjusted to the Mathieu group M 24 (Martin Liebeck, unpublished).For more on this comment, see [17, p. 116].See also Theorem 4.2.
Let us close this section by recording the following theorem.
Theorem 2.3 (Maróti [34]).Let G be a primitive permutation group of degree n.Then at least one of the following holds.
(1) G is a subgroup of S m S r containing (A m ) r , where the action of S m is on k-element subsets of {1, . . ., m} and the wreath product has the product action of degree n = m k r .

Minimal degree.
One of the classical problems in the theory of permutation groups was to classify the permutation groups whose minimal degree is small.Let G be a primitive permutation group of degree n.If G contains a transposition or a 3-cycle, then G must be S n or A n .This means that, if G does not contain A n , then its minimal degree μ(G) is at least 4.There are many classical results on minimal degrees of primitive permutation groups.For example, see Bochert [7], Jordan [26], and Manning [33].A summary can be found in Wielandt's book [41].See also [19].
The best result on the minimal degree of a primitive permutation group obtained prior to the classification of finite simple groups is due to Babai [1,Theorem 6.14].He claims that this is the central result of his paper [1].

Theorem 3.1 (Babai [1]). If G is a primitive permutation group of degree n and not containing
The following stronger bound was proved using the classification of finite simple groups.Theorem 3.2 (Liebeck [28]).Let G be a primitive permutation group of degree n.At least one of the following holds.
(1) G is a subgroup of S m S r containing (A m ) r , where the action of S m is on k-element subsets of {1, . . ., m} and the wreath product has the product action of degree n = m k r .
Liebeck and Saxl [32] improved the bound μ(G) > n/(9 log n) in part (2) of Theorem 3.2 to μ(G) ≥ n/3.This result is deduced from the stronger result [32, Theorem 6.1] where the n/3 is replaced by n/2 at the cost of further exceptions in part (1) of Theorem 3.2.It is noted in [32, p. 268] that in all cases the minimal degrees of the groups S m S r in Theorem 3.2 are realized by a transposition in one of the factors S m of the base group.A useful consequence [32, Corollary 3] of these results is an improvement of Babai's general bound (Theorem 3.1) to μ(G) > 2( √ n − 1).Later Guralnick and Magaard [22] classified all primitive permutation groups of degree n with μ(G) ≤ n/2.All examples are essentially variants on alternating or symmetric groups acting on the set of subsets of some cardinality k or from orthogonal groups over the field of two elements acting on some collection of 1-spaces or hyperplanes.
The strongest theorem to date on the minimal degree of a primitive permutation group is due to Burness and Guralnick [9,Theorem 4].[9]).Let G be a primitive permutation group acting on a finite set Ω of size n.Let a point-stabilizer be H.Either μ(G) ≥ 2n/3 or one of the following holds (up to permutation isomorphism).

Theorem 3.3 (Burness and Guralnick
( (6) G ≤ L S r is a product type primitive group with its product action on Ω = Γ r where r ≥ 2 and L ≤ Sym(Γ) is one of the almost simple groups in (1)-(4).
4. Base size.The minimal size of a base of a primitive permutation group has been much investigated.Already in the nineteenth century Bochert [6] showed that b(G) ≤ n/2 for a primitive permutation group G of degree n not containing A n .This bound was substantially improved by Babai to b(G) < 4 √ n ln n, for uniprimitive groups G, in [1], and to the estimate b(G) < 2 c √ log n for a universal constant c, for doubly transitive groups G not containing A n , in [2].The latter bound was improved by Pyber [37] to b(G) < c(log n) 2 where c is a universal constant.These estimates are elementary in the sense that their proofs do not require the classification of finite simple groups.
Using the classification, Liebeck [28] classified all primitive permutation groups G of degree n with b(G) ≥ 9 log n.Theorem 4.1 (Liebeck [28]).Let G be a primitive permutation group of degree n.At least one of the following holds.(2) b(G) < 9 log n.
In 1981, Babai conjectured (see [38, p. 207]) that there is a function f : N → N such that any primitive group that has no alternating or classical composition factor of degree or dimension greater than d has base size less than f (d).An important special case was a famous theorem of Seress [39]: if G is a (finite) solvable primitive permutation group then b(G) ≤ 4. Babai's conjecture was proved by Gluck et al. [21] with f a quadratic function and improved to a linear function f by Liebeck and Shalev [29].
In order to state a second conjecture on base sizes of primitive permutation groups, we return to Theorem 2.2 (and to the comment that follows it).If G is a primitive permutation group satisfying part (1) or part (2) of Theorem 2.2, then (the action of) G is called standard.Otherwise the action is said to be nonstandard.A well-known conjecture of Cameron and Kantor (see [16,18]) asserted that there exists an absolute constant c such that b(G) ≤ c for all almost simple primitive permutation groups G in nonstandard actions.Referring to c, Cameron wrote (see [17, p. 122]), 'Probably this constant is 7, and the extreme case is the Mathieu group M 24 '.The Cameron-Kantor conjecture was proved by Liebeck and Shalev [29] and in the strong form with c = 7 by Burness [8] and Burness et al. [10,12,13].Theorem 4.2 (Burness et al. [12,13]).If G is an almost simple primitive permutation group in nonstandard action, then b(G) ≤ 7, with equality if and only if G is the Mathieu group M 24 in its natural action of degree 24.
2 2 with n = 22 and μ(G) = 14.(4) G is an almost simple classical group in a subspace action and the few possibilities are listed in [9, Table 2].(5) G = V : H is an affine group with unique minimal normal subgroup V = (C 2 ) d and H ≤ GL d (2) contains a transvection and μ(G) = 2 d−1 = n/2.

( 1 )
G is a subgroup of S m S r containing (A m ) r , where the action of S m is on k-element subsets of {1, . . ., m} and the wreath product has the product action of degree n = m k r .