Finite semiprimitive permutation groups of rank 3

Finite Permutation Groups

Quasiprimitivity: structure and combinatorial applications

The theory of vertex-transitive graphs has developed in parallel with the theory of transitive permutation groups. The chapter explores the way the two theories have influenced each other. Examples are drawn from the enumeration of vertex-transitive graphs of small order, the classification problem for finite distance transitive graphs, and the investigations of finite 2-arc transitive graphs, finite primitive and quasiprimitive permutation groups, and finite locally primitive graphs. The nature of the group theoretic techniques used range from elementary ones to some involving the finite simple group classification. In particular the theorem of O’Nan and Scott for finite primitive permutation groups, and a generalisation of it for finite quasiprimitive permutation groups is discussed.

On graph-restrictive permutation groups

Suppose G is a transitive permutation group on a finite set \(\mit\Omega \) of n points and let p be a prime divisor of \(|G|\). The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted mp (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that \(m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega |\) holds, if \({\rm Alt}(\mit\Omega ) \nleqq G \).¶Also all groups G (and prime divisors p of \(|G|\)) for which \(m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega |\) are identified.

Various lattices of subgroups of a finite transitive permutation group G can be used to define a set of ‘basic’ permutation groups associated with G that are analogues of composition factors for abstract finite groups. In particular G can be embedded in an iterated wreath product of a chain of its associated basic permutation groups. The basic permutation groups corresponding to the lattice L of all subgroups of G containing a given point stabiliser are a set of primitive permutation groups. We introduce two new subgroup lattices contained in L, called the seminormal subgroup lattice and the subnormal subgroup lattice. For these lattices the basic permutation groups are quasiprimitive and innately transitive groups, respectively.

A finite permutation group [Formula: see text] is called [Formula: see text]-closed if [Formula: see text] is the largest subgroup of [Formula: see text] which leaves invariant each of the [Formula: see text]-orbits for the induced action on [Formula: see text]. Introduced by Wielandt in 1969, the concept of [Formula: see text]-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total [Formula: see text]-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group [Formula: see text] is said to be totally [Formula: see text]-closed if [Formula: see text] is [Formula: see text]-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally [Formula: see text]-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly [Formula: see text] totally [Formula: see text]-closed finite nonabelian simple groups: the Janko groups [Formula: see text], [Formula: see text] and [Formula: see text], together with [Formula: see text], [Formula: see text] and the Monster [Formula: see text]. Moreover, if a finite totally [Formula: see text]-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely [Formula: see text] examples. In the course of obtaining this classification, we develop a general framework for studying [Formula: see text]-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.

Fast recognition of doubly transitive groups

According to P. Hall, a subgroup \(H\) of a finite group \(G\) is called pronormal in \(G\) if, for any element \(g\) of \(G\), the subgroups \(H\) and \(H^{g}\) are conjugate in \(\langle H,H^{g}\rangle\). The simplest examples of pronormal subgroups of finite groups are normal subgroups, maximal subgroups, and Sylow subgroups. Pronormal subgroups of finite groups were studied by a number of authors. For example, Legovini (1981) studied finite groups in which every subgroup is subnormal or pronormal. Later, Li and Zhang (2013) described the structure of a finite group \(G\) in which, for a second maximal subgroup \(H\), its index in \(\langle H,H^{g}\rangle\) does not contain squares for any \(g\) from \(G\). A number of papers by Kondrat’ev, Maslova, Revin, and Vdovin (2012–2019) are devoted to studying the pronormality of subgroups in a finite simple nonabelian group and, in particular, the existence of a nonpronormal subgroup of odd index in a finite simple nonabelian group. In The Kourovka Notebook, the author formulated Question 19.109 on the equivalence in a finite simple nonabelian group of the condition of pronormality of its second maximal subgroups and the condition of Hallness of its maximal subgroups. Tyutyanov gave a counterexample \(L_{2}(2^{11})\) to this question. In the present paper, we provide necessary and sufficient conditions for the pronormality of second maximal subgroups in the group \(L_{2}(q)\). In addition, for \(q\leq 11\), we find the finite almost simple groups with socle \(L_{2}(q)\) in which all second maximal subgroups are pronormal.

A theory of semiprimitive groups

In January 1969, Peter M. Neumann wrote a paper entitled “Primitive permutation groups of degree 3p”. The main theorem placed restrictions on the parameters of a primitive but not 2-transitive permutation group of degree three times a prime. The paper was never published, and the results have been superseded by stronger theorems depending on the classification of the finite simple groups, for example a classification of primitive groups of odd degree.However, there are further reasons for being interested in this paper. First, it was written at a time when combinatorial techniques were being introduced into the theory of finite permutation groups, and the paper gives a very good summary and application of these techniques. Second, like its predecessor by Helmut Wielandt on primitive groups of degree 2p, it can be re-interpreted as a combinatorial result concerning association schemes whose common eigenspaces have dimensions of a rather limited form. This result uses neither the primality of p nor the existence of a permutation group related to the combinatorial structure. We extract these results and give details of the related combinatorics.

Let $G = X \wr H$ be the wreath product of a nontrivial finite group X with k conjugacy classes and a transitive permutation group H of degree n acting on the set of n direct factors of X n . If H is semiprimitive, then $k(G) \leq k^n$ for every sufficiently large n or k . This result solves a case of the non-coprime k ( GV ) problem and provides an affirmative answer to a question of Garzoni and Gill for semiprimitive permutation groups. The proof does not require the classification of finite simple groups.

Let G be a transitive permutation group on a finite set of size at least 2. By a well known theorem of Fein, Kantor and Schacher, G contains a derangement of prime power order. In this paper, we study the finite primitive permutation groups with the extremal property that the order of every derangement is an r-power, for some fixed prime r. First we show that these groups are either almost simple or affine, and we determine all the almost simple groups with this property. We also prove that an affine group G has this property if and only if every two-point stabilizer is an r-group. Here the structure of G has been extensively studied in work of Guralnick and Wiegand on the multiplicative structure of Galois field extensions, and in later work of Fleischmann, Lempken and Tiep on \({r'}\)-semiregular pairs.

There is a familiar construction with two finite, transitive permutation groups as input and a finite, transitive permutation group, called their wreath product, as output. The corresponding ‘imprimitive wreath decomposition’ concept is the first subject of this paper. A formal definition is adopted and an overview obtained for all such decompositions of any given finite, transitive group. The result may be heuristically expressed as follows, exploiting the associative nature of the construction. Each finite transitive permutation group may be written, essentially uniquely, as the wreath product of a sequence of wreath-indecomposable groups, amid the two-factor wreath decompositions of the group are precisely those which one obtains by bracketing this many-factor decomposition.If both input groups are nontrivial, the output above is always imprimitive. A similar construction gives a primitive output, called the wreath product in product action, provided the first input group is primitive and not regular. The second subject of the paper is the ‘product action wreath decomposition’ concept dual to this. An analogue of the result stated above is established for primitive groups with nonabelian socle.Given a primitive subgroup G with non-regular socle in some symmetric group S, how many subgroups W of S which contain G and have the same socle, are wreath products in product action? The third part of the paper outlines an algorithm which reduces this count to questions about permutation groups whose degrees are very much smaller than that of G.

Let $G$ be a permutation group on a set $\Omega$. Then for each $g\in G$, we define the movement of $g$, denoted by $\move(g)$, the maximal cardinality $|\Delta^{g}\backslash \Delta|$ of $\Delta^{g}\backslash \Delta$ over all subsets $\Delta$ of $\Omega$. And the movement of $G$ is defined as the maximum of $\move(g)$ over all $g\in G$, denoted by $\move(G)$. A permutation group $G$ is said to have bounded movement if it has movement bounded by some positive integer $m$, that is $\move(G)\leq m$. In this paper, we consider the finite transitive permutation groups $G$ with movement $\move(G)=m$ for some positive integer $m>4$, where $G$ is not a $2$-group but in which every non-identity element has the movement $m$ or $m-4$, and there is at least one non-identity element that has the movement $m-4$. We give a characterization for elements of $G$ in Theorem\ref{thm-1}. Further, we apply Theorem \ref{thm-1} to character transitive permutation group $G$ in Theorem \ref{thm-2}. These results give a partial answer to the open problem posed by the authors in 2024.