Abstract
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.
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1 Introduction
It has been 40 years since Martin W. Liebeck’s article “On minimal degrees and base sizes of primitive permutation groups” was published in Archiv der Mathematik. We take the opportunity to give a short survey of selected topics on the effects of this influential paper.
Let G be a finite permutation group acting on a set \(\Omega .\) This is said to be transitive if for every \(\alpha \) and \(\beta \) in \(\Omega ,\) there is a permutation g in G such that \(\alpha g = \beta .\) A nonempty subset \(\Delta \) of \(\Omega \) is called a block for G if for every \(g \in G,\) the set \(\Delta g \cap \Delta \) is either equal to \(\Delta \) or is the empty set. A block is trivial if it has size 1 or if it is \(\Omega .\) If there is no nontrivial block for a transitive permutation group G, then G is called a primitive permutation group. For an element \(\alpha \in \Omega ,\) let \(G_{\alpha }\) denote the point-stabilizer of \(\alpha \) in G defined to be the subgroup of G consisting of those elements g such that \(\alpha g = \alpha .\) (A transitive permutation group G is primitive if and only if \(G_{\alpha }\) is a maximal subgroup in G for some (any) \(\alpha \in \Omega .\))
Let \(\mu (G)\) be the smallest number of elements of \(\Omega \) moved by any non-identity permutation in G. This invariant is called the minimal degree of G. A nonempty subset \(\Delta \) of \(\Omega \) is called a base for G if \(\cap _{\alpha \in \Delta } G_{\alpha } = 1.\) The minimal size of a base for G is denoted by b(G). The third invariant which is discussed in this paper is the order |G| of G.
The degree of G is defined to be \(|\Omega |,\) which we denote by n. Knowledge of the minimal size b(G) of a base for G provides lower and upper bounds for |G|. It is easy to see that \(2^{b(G)} \le |G| \le n(n-1) \cdots (n-b(G)+1) \le n^{b(G)}.\) The relationship between the minimal size of a base for G and the minimal degree of G is the following. If G is a transitive permutation group, then \(n \le b(G) \mu (G)\) (see [19, Exercise 3.3.7]).
Martin Liebeck’s article [28] contains three main results about a finite primitive permutation group G and these concern the invariants |G|, \(\mu (G),\) and b(G). We restate these groundbreaking theorems and discuss the three invariants in three separate sections. The survey could be considered as an account of existing results in the selected areas. Liebeck’s paper [28] appeared soon after the classification of finite simple groups was announced. We give a brief history, with some relevant references for the interested reader, of work done on these topics before 1984. We then describe (in our point of view) the most direct influence of Liebeck’s theorems. Many references are given on our selected topics. The reader is advised to consult these for more information and motivation. There is new progress (in different directions) and there are new questions and problems. These are not discussed here since we tried to keep the survey as concise as possible, however we hope that the reader will become interested and can get acquainted with the most recent topics influenced by Martin Liebeck’s paper [28].
2 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 \((S_n: G) \ge [{1 \over 2}(n+1)]!.\) 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 \(e^{4 \sqrt{n} \ln ^2 n}\) estimate for uniprimitive (primitive but not doubly transitive) groups. (A permutation group G acting on a set \(\Omega \) is doubly transitive if for any two tuples \((\alpha _{1}, \beta _{1})\) and \((\alpha _{2},\beta _{2})\) from \(\Omega \times \Omega \) such that \(\alpha _{1} \not = \beta _{1}\) and \(\alpha _{2} \not = \beta _{2},\) there exists a permutation \(g \in G\) with \(\alpha _{1}g = \alpha _{2}\) and \(\beta _{1}g = \beta _{2}.\) Doubly transitive groups are primitive.) For the orders of doubly transitive groups not containing \(A_n,\) Babai [2] obtained the bound \(\exp ( \exp (c \sqrt{\ln n}))\) for some universal constant c. This was improved by Pyber [37] to an \(n^{c \cdot {\ln ^2 n}}\) bound by an elementary argument (using some ideas of [2]). Apart from \(\textrm{O}(\ln n)\) 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 \(\Gamma ,\) then the wreath product \(H \wr S_{r}\) acts in a natural way on the set \(\Gamma ^{r}.\) 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 \({\textrm{Aut}}(T) \wr S_{r},\) 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 \(T = {\textrm{PSL}}_{d}(q)\) 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.
-
(3)
\(|G| \le n^{c \ln \ln n}.\)
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 \(\{1, \ldots , m \}\) or on partitions of \(\{ 1, \ldots , m \}\) into a subsets of size b, where \(ab=m,\) \(a > 1,\) \(b > 1,\) \(n = \left( {\begin{array}{c}m\\ k\end{array}}\right) \) or \(m!/(b!)^{a}a!,\) respectively.
-
(2)
T is a classical simple group acting on an orbit of subspaces of the natural module, or (in the case \(T = {\textrm{PSL}}_{d}(q)\) where d is an integer and q is a prime power) pairs of subspaces of complementary dimensions.
-
(3)
\(|G| < n^9.\)
The bound \(|G| < n^{9}\) in part (3) of Theorem 2.2 may be replaced by \(|G| \le n^{c}\) where \(c = 6.077 948 094\) is adjusted to the Mathieu group \({\textrm{M}}_{24}\) (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 \(S_m \wr \ S_r\) containing \((A_m)^r,\) where the action of \(S_m\) is on k-element subsets of \(\{ 1, \ldots , m \}\) and the wreath product has the product action of degree \(n= {m \atopwithdelims ()k}^r.\)
-
(2)
\(G = {\textrm{M}}_{11},\) \({\textrm{M}}_{12},\) \({\textrm{M}}_{23},\) or \({\textrm{M}}_{24}\) with their 4-transitive actions.
-
(3)
\(|G| \le n \cdot \prod _{i=0}^{[\log _2 n] -1} (n-2^i) < n^{1+[{\log _2 n}]}.\)
3 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 \(\mu (G)\) 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 \(A_{n},\) then \(\mu (G) > (\sqrt{n}-1)/2.\)
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 \(S_{m} \wr S_{r}\) containing \({(A_{m})}^{r},\) where the action of \(S_m\) is on k-element subsets of \(\{ 1, \ldots , m \}\) and the wreath product has the product action of degree \(n = {\left( {\begin{array}{c}m\\ k\end{array}}\right) }^{r}.\)
-
(2)
\(\mu (G) > n/(9 \log n).\)
Liebeck and Saxl [32] improved the bound \(\mu (G) > n/(9 \log n)\) in part (2) of Theorem 3.2 to \(\mu (G) \ge n/3.\) 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 \(S_{m} \wr S_r\) 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 \(\mu (G) > 2 (\sqrt{n}-1).\) Later Guralnick and Magaard [22] classified all primitive permutation groups of degree n with \(\mu (G) \le n/2.\) 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].
Theorem 3.3
(Burness and Guralnick [9]). Let G be a primitive permutation group acting on a finite set \(\Omega \) of size n. Let a point-stabilizer be H. Either \(\mu (G) \ge 2n/3\) or one of the following holds (up to permutation isomorphism).
-
(1)
\(G = S_{m}\) or \(A_{m}\) acting on k-element subsets of \(\{ 1, \ldots , m \}\) with \(1 \le k < m/2.\)
-
(2)
\(G = S_{m},\) \(H = S_{m/2} \wr S_{2},\) and \(\mu (G) = (1+ 1/(m-1)) (n/2).\)
-
(3)
\(G = {\textrm{M}}_{22}:2\) or \(G = {\textrm{L}}_{3}(4).2_{2}\) with \(n=22\) and \(\mu (G) = 14.\)
-
(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 \(H \le {\textrm{GL}}_{d}(2)\) contains a transvection and \(\mu (G) = 2^{d-1} = n/2.\)
-
(6)
\(G \le L \wr S_{r}\) is a product type primitive group with its product action on \(\Omega = \Gamma ^{r}\) where \(r \ge 2\) and \(L \le {\textrm{Sym}}(\Gamma )\) 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 \(b(G) \le n/2\) for a primitive permutation group G of degree n not containing \(A_{n}.\) This bound was substantially improved by Babai to \(b(G) < 4 \sqrt{n} \ln n,\) for uniprimitive groups G, in [1], and to the estimate \(b(G) < 2^{c \sqrt{\log n}}\) 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 \(b(G) \ge 9 \log n.\)
Theorem 4.1
(Liebeck [28]). Let G be a primitive permutation group of degree n. At least one of the following holds.
-
(1)
G is a subgroup of \(S_{m} \wr S_{r}\) containing \({(A_{m})}^{r},\) where the action of \(S_m\) is on k-element subsets of \(\{ 1, \ldots , m \}\) and the wreath product has the product action of degree \(n = {\left( {\begin{array}{c}m\\ k\end{array}}\right) }^{r}.\)
-
(2)
\(b(G) < 9 \log n.\)
In 1981, Babai conjectured (see [38, p. 207]) that there is a function \(f: {\mathbb {N}} \rightarrow {\mathbb {N}}\) 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 \(b(G) \le 4.\) 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 \(b(G) \le c\) 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 \({\textrm{M}}_{24}\)’. 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 \(b(G) \le 7,\) with equality if and only if G is the Mathieu group \({\textrm{M}}_{24}\) in its natural action of degree 24.
For primitive permutation groups G in standard actions, the minimal base size b(G) cannot be bounded from above by an absolute constant. This is because, in general, the orders of the permutation groups in standard actions are not bounded from above by a fixed polynomial function of their degree and thus b(G) is unbounded by the observation that \(b(H) > \log |H| / \log n\) for any finite permutation group H of degree n. Bounds (and formulas) for b(G) for groups G satisfying part (1) of Theorem 2.2 were obtained by Halasi [23], Bailey and Cameron [3, 5, 14] and James [25]. For bounds and calculations for b(G) in case G satisfies part (2) of Theorem 2.2, see [24, 35], and [11, Section 4].
A well-known conjecture of Pyber [38, p. 207] stated that there exists a universal constant c such that \(b(G) < c (\log |G| / \log n)\) for all primitive permutation groups G of degree n. For a collection of results towards this conjecture, see the survey article [31]. See also the important paper [30] of Liebeck and Shalev. The proof of Pyber’s conjecture was completed by Duyan, Halasi, and the author [20]. It was shown in [20] that there exists a universal constant c such that, for every primitive permutation group G of degree n, we have \(b(G) < 45 (\log |G| / \log n) + c.\) The theorem was made effective by Halasi et al. [24] (where the multiplicative constant 2 is best possible).
Theorem 4.3
(Halasi et al. [24]). If G is a primitive permutation group of degree n, then \(b(G) \le 2 (\log |G| / \log n) + 24.\)
We continue this section with a beautiful improvement of Theorem 4.1 due to Moscatiello and Roney-Dougal [35]. The next theorem should also be compared with Theorem 2.3.
Theorem 4.4
(Moscatiello and Roney-Dougal [35]). Let G be a primitive permutation group of degree n and not satisfying part (1) of Theorem 4.1. Then \(b(G) \ge 1 + \log n\) if and only if G is one of the following.
-
(1)
A subgroup of \({\textrm{AGL}}_{d}(2)\) for some integer d and \(b(G) = 1+d = 1+\log n.\)
-
(2)
The group \({\textrm{Sp}}_{d}(2),\) acting on the cosets of \({\textrm{GO}}^{-}_{d}(2)\) with \(d \ge 4,\) in which case \(1 + \log n < b(G) = 1 + \lceil \log n \rceil .\)
-
(3)
A Mathieu group \({\textrm{M}}_{n}\) in its natural permutation representation with n in the set \(\{ 12, 23, 24 \}.\) If \(n = 12\) or 23, then \(b(G) = 1 + \lceil \log n \rceil ,\) while if \(n = 24,\) then \(b(G) = 7 > 1 + \lceil \log n \rceil .\)
Concerning the groups satisfying part (1) of Theorem 4.1, we remark that if G is a primitive permutation group of degree n not containing \(A_{n},\) then \(b(G) \le \max \{\sqrt{n}, \ 25\}\) by [24, Corollary 1.3].
The base size has important applications in computational group theory. See for example [40]. Blaha [4] proved that the problem of computing a minimal base for a permutation group is NP-hard. However, an irredundant base (see the next paragraph) can be computed in polynomial time (see [40]).
Let G be a permutation group acting on a finite set \(\Omega .\) A nonempty subset \(\Delta \) of \(\Omega \) is called an irredundant base for G if it is a base for G but no proper subset of \(\Delta \) is a base for G. The maximal size of an irredundant base for G (acting on \(\Omega \)) is denoted by I(G). Another nice improvement of Theorem 4.1 was obtained by Kelsey and Roney-Dougal [27]. Let G be a primitive permutation group of degree n not satisfying part (1) of Theorem 4.1. Then \(I(G) < 5 \log n.\) It follows from this and the previous paragraph that a base for G of size at most \(5 \log n\) can be constructed in polynomial time (see [27]).
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Maróti, A. Minimal degree, base size, order: selected topics on primitive permutation groups. Arch. Math. 121, 485–493 (2023). https://doi.org/10.1007/s00013-023-01905-7
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DOI: https://doi.org/10.1007/s00013-023-01905-7