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Fix a positive integer d and let Γ d be the class of finite groups without sections isomorphic to the alternating group A d. The groups in Γ d were studied by Babai, Cameron and Pálfy in the 1980s and they determined bounds on the order of a primitive permutation group with this property, which have found a wide range of applications. Subsequently, results on the base sizes of such groups were also obtained. In this paper we replace the structural conditions on the group by restrictions on its point stabilizers, and we obtain similar, and sometimes stronger conclusions. For example, we prove that there is a linear function f such that the base size of any finite primitive group with point stabilizers in Γ d is at most f (d). This generalizes a recent result of the first author on primitive groups with solvable point stabilizers. For non-affine primitive groups we obtain stronger results, assuming only that stabilizers of c points lie in Γ d. We also show that if G is any permutation group of degree n whose c -point stabilizers lie in Γ d , then | G | ⩽ ((1 + o c (1)) d / e) n − 1. This asymptotically extends and improves a d n − 1 upper bound on | G | obtained by Babai, Cameron and Pálfy assuming G ∈ Γ d. [ABSTRACT FROM AUTHOR]

Dedicated to the memory of Jan Saxl   Let |$G$| be a finite primitive permutation group on a set |$\Omega $| with nontrivial point stabilizer |$G_{\alpha }$|⁠. We say that |$G$| is extremely primitive if |$G_{\alpha }$| acts primitively on each of its orbits in |$\Omega \setminus \{\alpha \}$|⁠. These groups arise naturally in several different contexts, and their study can be traced back to work of Manning in the 1920s. In this paper, we determine the almost simple extremely primitive groups with socle an exceptional group of Lie type. By combining this result with earlier work of Burness, Praeger, and Seress, this completes the classification of the almost simple extremely primitive groups. Moreover, in view of results by Mann, Praeger, and Seress, our main theorem gives a complete classification of all finite extremely primitive groups, up to finitely many affine exceptions (and it is conjectured that there are no exceptions). Along the way, we also establish several new results on base sizes for primitive actions of exceptional groups, which may be of independent interest. [ABSTRACT FROM AUTHOR]

Let G be a nontrivial transitive permutation group on a finite set Ω. An element of G is said to be a derangement if it has no fixed points on Ω. From the orbit counting lemma, it follows that G contains a derangement, and in fact G contains a derangement of prime power order by a theorem of Fein, Kantor and Schacher. However, there are groups with no derangements of prime order; these are the so-called elusive groups and they have been widely studied in recent years. Extending this notion, we say that G is almost elusive if it contains a unique conjugacy class of derangements of prime order. In this paper we first prove that every quasiprimitive almost elusive group is either almost simple or 2-transitive of affine type. We then classify all the almost elusive groups that are almost simple and primitive with socle an alternating group, a sporadic group, or a rank one group of Lie type. [ABSTRACT FROM AUTHOR]

Let G be a permutation group on a set Ω. A subset of Ω is a base for G if its pointwise stabiliser in G is trivial. In this paper we introduce and study an associated graph Σ(G), which we call the Saxl graph of G. The vertices of Σ(G) are the points of Ω, and two vertices are adjacent if they form a base for G. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of Σ(G) for a finite transitive group G , as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For instance, we conjecture that if G is a primitive group with a base of size 2, then the diameter of Σ(G) is at most 2. Using a probabilistic approach, we establish the conjecture for some families of almost simple groups. For example, the conjecture holds when G = Sn or An (with n > 12) and the point stabiliser of G is a primitive subgroup. In contrast, we can construct imprimitive groups whose Saxl graph is disconnected with arbitrarily many connected components, or connected with arbitrarily large diameter. [ABSTRACT FROM AUTHOR]

Abstract Let G ⩽ Sym (Ω) be a finite almost simple primitive permutation group with socle G 0. A subset of Ω is a base for G if its pointwise stabilizer is trivial; the base size of G , denoted b (G) , is the minimal size of a base. We say that G is standard if G 0 = A n and Ω is an orbit of subsets or partitions of { 1 , ... , n } , or if G 0 is a classical group and Ω is an orbit of subspaces (or pairs of subspaces) of the natural module for G 0. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have b (G) ⩽ 7 for every non-standard group G , with equality if and only if G is the Mathieu group M 24 in its natural action on 24 points. In this paper, we extend this result by classifying the non-standard groups with b (G) = 6. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type. [ABSTRACT FROM AUTHOR]

Let G ⩽ Sym (Ω) be a finite permutation group and recall that the base size of G is the minimal size of a subset of Ω with trivial pointwise stabiliser. There is an extensive literature on base sizes for primitive groups, but there are very few results for primitive groups of product type. In this paper, we initiate a systematic study of bases in this setting. Our first main result determines the base size of every product type primitive group of the form L ≀ P ⩽ Sym (Ω) with soluble point stabilisers, where Ω = Γ k , L ⩽ Sym (Γ) and P ⩽ S k is transitive. This extends recent work of Burness on almost simple primitive groups. We also obtain an expression for the number of regular suborbits of any product type group of the form L ≀ P and we classify the groups with a unique regular suborbit under the assumption that P is primitive, which involves extending earlier results due to Seress and Dolfi. We present applications on the Saxl graphs of base-two product type groups and we conclude by establishing several new results on base sizes for general product type primitive groups. [ABSTRACT FROM AUTHOR]

Let G be a finite primitive permutation group on a set Ω and recall that the fixed point ratio of an element x ∈ G , denoted fpr (x) , is the proportion of points in Ω fixed by x. Fixed point ratios in this setting have been studied for many decades, finding a wide range of applications. In this paper, we are interested in comparing fpr (x) with the order of x. Our main theorem provides a classification of the triples (G , Ω , x) as above with the property that x has prime order r and fpr (x) > 1 / (r + 1). There are several applications. Firstly, we extend earlier work of Guralnick and Magaard by determining the primitive permutation groups of degree m with minimal degree at most 2 m / 3. Secondly, our main result plays a key role in recent joint work with Moretó and Navarro on the commuting probability of p -elements in finite groups. Finally, we use our main theorem to investigate the minimal index of a primitive permutation group, which allows us to answer a question of Bhargava. [ABSTRACT FROM AUTHOR]

Let G be a permutation group on a finite set Q. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. The base size of G, denoted b(G), is the smallest size of a base. A well-known conjecture of Pyber from the early 1990s asserts that there exists an absolute constant c such that b(G) ≤ c log|G|/log n for any primitive permutation group G of degree n. Several special cases have been verified in recent years, including the almost simple and diagonal cases. In this paper, we prove Pyber's conjecture for all non-affine primitive groups. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finite group and let be the minimal number of generators for G. It is well known that for all (non-abelian) finite simple groups. We prove that for any maximal subgroup H of a finite simple group, and that this bound is best possible. We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant. We then apply our results to the study of permutation groups. In particular we show that if G is a finite primitive permutation group with point stabilizer H, then . [Copyright &y& Elsevier]

A non-regular primitive permutation group is said to be extremely primitive if a point stabilizer acts primitively on each of its orbits. By a theorem of Mann and the second and third authors, every finite extremely primitive group is either almost simple or of affine type. In a recent paper, we classified the extremely primitive almost simple classical groups, and in this note we determine the examples with a sporadic or alternating socle. We obtain two infinite families for An (or Sn); they comprise the natural 2-primitive action of n points, plus the action on partitions of {1, … , n} into subsets of size n/2 (with n/2 odd). There are twenty examples for sporadic groups, including the rank 6 representation of Co2 on the cosets of McL. [ABSTRACT FROM PUBLISHER]

Abstract: A primitive permutation group is said to be extremely primitive if it is not regular and a point stabilizer acts primitively on each of its orbits. By a theorem of Mann and the second and third authors, every finite extremely primitive group is either almost simple or of affine type. In this paper, we determine the examples in the case of almost simple classical groups. They comprise the -transitive actions of and its extensions of degree , and of of degrees , together with the -transitive actions of on cosets of , with a Fermat prime. In addition to these three families, there are four individual examples. [Copyright &y& Elsevier]

Abstract: Let G be a transitive permutation group on a finite set Ω of size at least 2. An element of G is a derangement if it has no fixed points on Ω. Let r be a prime divisor of . We say that G is r-elusive if it does not contain a derangement of order r, and strongly r-elusive if it does not contain one of r-power order. In this note we determine the r-elusive and strongly r-elusive primitive actions of almost simple groups with socle an alternating or sporadic group. [Copyright &y& Elsevier]

A base of a permutation group G on a set Ω is a subset B of Ω such that the pointwise stabilizer of B in G is trivial. The base size of G, denoted by b(G), is the minimal cardinality of a base. Let G = Sn or An acting primitively on a set with point stabilizer H. In this note, we prove that if H acts primitively on {1, …, n}, and does not contain An, then b(G) = 2 for all n ≥ 13. Combined with a theorem of James, this completes the classification of primitive actions of alternating and symmetric groups which admit a base of size 2. [ABSTRACT FROM AUTHOR]

Let G be a finite almost simple group with socle G 0. A (nontrivial) factorization of G is an expression of the form G = H K , where the factors H and K are core-free subgroups. There is an extensive literature on factorizations of almost simple groups, with important applications in permutation group theory and algebraic graph theory. In a recent paper, Xia and the second-named author describe the factorizations of almost simple groups with a solvable factor H. Several infinite families arise in the context of classical groups and in each case a solvable subgroup of G 0 containing H ∩ G 0 is identified. Building on this earlier work, in this paper we compute a sharp lower bound on the order of a solvable factor of every almost simple group and we determine the exact factorizations with a solvable factor. As an application, we describe the finite primitive permutation groups with a nilpotent regular subgroup, extending classical results of Burnside and Schur on cyclic regular subgroups, and more recent work of Li in the abelian case. [ABSTRACT FROM AUTHOR]