Finite Permutation Groups

Journal of the London Mathematical Society, 1973

In a series of papers [3, 4 and 5] on insoluble (transitive) permutation groups of degree p = 2q +1, where p and q are primes, N. Ito has shown that, apart from a small number of exceptions, such a group must be at least quadruply transitive. One of the results which he uses is that an insoluble group of degree p = 2q +1 which is not doubly primitive must be isomorphic to PSL (3, 2) with p = 7. This result is due to H. Wielandt, and ltd gives a proof in [3]. It is quite easy to extend this proof to give the following result: a doubly transitive group of degree 2q + l, where q is prime, which is not doubly primitive, is either sharply doubly transitive or a group of automorphisms of a block design with A = 1 and k = 3. Our notation for the parameters of a block design, v, b, k, r, X, is standard; see [9]. In this paper we shall prove two results about doubly transitive but not doubly primitive groups which resemble the two results mentioned above.

Monatshefte für Mathematik, 2011

If V is a (possibly infinite) set, G a permutation group on V , v ∈ V , and Ω is an orbit of the stabiliser G v , let G Ω v denote the permutation group induced by the action of G v on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G v and G Ω v. Among other things, we prove that if G is transitive, G v is finite, and the N-orbital {(v g , u g) | u ∈ Ω, g ∈ N (G)} is strongly connected (when viewed as a digraph on V), then every simple homomorphic image of a subgroup of G v is also a homomorphic image of a subgroup of G Ω v. To demonstrate that the assumption on finiteness of G v cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G v is isomorphic to the modular group PSL(2, Z) ∼ = C 2 * C 3 .

arXiv (Cornell University), 2023

A transitive permutation group G on a finite set Ω is said to be pre-primitive if every G-invariant partition of Ω is the orbit partition of a subgroup of G. It follows that pre-primitivity and quasiprimitivity are logically independent (there are groups satisfying one but not the other) and their conjunction is equivalent to primitivity. Indeed, part of the motivation for studying pre-primitivity is to investigate the gap between primitivity and quasiprimitivity. We investigate the pre-primitivity of various classes of transitive groups including groups with regular normal subgroups, direct and wreath products, and diagonal groups. In the course of this investigation, we describe all G-invariant partitions for various classes of permutation groups G. We also look briefly at conditions similarly related to other pairs of conditions, including transitivity and quasiprimitivity, k-homogeneity and k-transitivity, and primitivity and synchronization.

Annals of the New York Academy of Sciences, 1979

Journal of the London Mathematical Society, 1988

The concept of the 2-closure of a finite permutation group was introduced by Wielandt in [15]. We recall the definition: if G is a finite permutation group on a set ft, the 2-closure G <2) of G is the largest subgroup of Sym (Q) containing G which has the same orbits as G in the induced action on Q x Q. Wielandt obtains a number of properties of G <2> in [15], and there are some interesting applications in [12,13,15]. Clearly, if G is primitive of rank r on Q then so is G <2) , and G (2> is the largest subgroup of Sym (Q) containing G with this property. In particular, if G is 2-transitive on Q. then G (2> is the full symmetric group Sym (Q). It is natural to ask about the size of G (2) when G is simply primitive (that is, primitive but not 2-transitive). For nonabelian simple groups of order less than 10 6 this is one of the themes in [7]. In this paper we use the result of [9] to investigate G (2) whenever G is almost simple, that is, L o G < Aut L for some non-abelian simple group L. Our main result shows that if G is almost simple and simply primitive on ft, then G <2) is in general contained in the normalizer of L in Sym (Q). There are, however, some striking examples where this is not so. Apart from six 'sporadic' rank 3 examples which are probably well known, there are two infinite families with unbounded ranks. THEOREM 1. Let G be an almost simple group with L <i G ^ Aut L and L simple. Assume that G is simply primitive on a set ft of size n. Then G (2) is contained in the normalizer of L in Sym (Q) except in the following examples: (1) L = G 2 (q) with q^3,n = q\q z-\)/2 and the socle of G™ is Q 7 (#); (2) L = Cl 7 (q), n = ^3(^4-1)/(2, ?-1) and the socle of G™ is PQ£(tf); (3) six exceptional examples of rank 3 given in Table 1.

Bulletin of the London Mathematical Society, 1999

Let G be a transitive permutation group on a set Ω such that, for ω ? Ω, the stabiliser G ω induces on each of its orbits in ΩBoωq a primitive permutation group (possibly of degree 1). Let N be the normal closure of G ω in G. Then (Theorem 1) either N factorises as N l G ω G δ for some ω, δ ? Ω, or all unfaithful G ω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the G ω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω.

Discrete Mathematics, 2010

, the codewords of small weight in the dual code of the code of points and lines of Q(4, q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q + (5, q) and H(5, q 2 ), and we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces of the classical polar spaces. Furthermore, we investigate the codewords with the largest weights in these codes, where for q even and k sufficiently small, we determine the maximum weight and characterise the codewords of maximum weight. Moreover, we show that there exists an interval such that for every even number w in this interval, there is a codeword in the dual code of Q + (5, q), q even, with weight w and we show that there is an empty interval in the weight distribution of the dual of the code of Q(4, q), q even. To prove this, we show that a blocking set of Q(4, q), q even, of size q 2 + 1 + r, where 0 < r < (q + 4)/6, contains an ovoid of Q(4, q), improving on [5, Theorem 9].

Journal of the Australian Mathematical Society, 2016

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

2014

A linear group G on a finite vector space V, (that is, a subgroup of GL(V)) is called (1/2)-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the (1/2)-transitive linear groups. As a consequence we complete the determination of the finite (3/2)-transitive permutation groups -- the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the finite (k+1/2)-transitive permutation groups for integers k > 1.

2000

All known finite sharply 4-transitive permutation sets containing the identity are groups, namely S 4 , S 5 , A 6 and the Mathieu group of degree 11. We prove that a sharply 4-transitive permutation set on 11 elements containing the identity must necessarily be the Mathieu group of degree 11. The proof uses direct counting arguments. It is based on a combinatorial property of the involutions in the Mathieu group of degree 11 (which is established here) and on the uniqueness of the Minkowski planes of order 9 (which had been established before): the validity of both facts relies on computer calculations. A permutation set is said to be invertible if it contains the identity and if whenever it contains a permutation it also contains its inverse. In the geometric structure arising from an invertible permutation set at least one block-symmetry is an automorphism. The above result has the following consequences. i) A sharply 5-transitive permutation set on 12 elements containing the identity is necessarily the Mathieu group of degree 12. ii) There exists no sharply 6-transitive permutation set on 13 elements. For d ≥ 6 there exists no invertible sharply d-transitive permutation set on a finite set with at least d + 3 elements. iii) A finite invertible sharply d-transitive permutation set with d ≥ 4 is necessarily a group, that is either a symmetric group, an alternating group, the Mathieu group of degree 11 or the Mathieu group of degree 12.