Enumerating Transitive Finite Permutation Groups

Denote by f(n) the number of subgroups of the symmetric groupSym(n) of degree n, and by f_trans(n) the number of its transitivesubgroups. It was conjectured by Pyber [9] that almost all subgroupsof Sym(n) are not transitive, that is, f_trans(n)/f(n) tendsto 0 when n tends to infinity. It is still an open questionwhether or not this conjecture is true. The difficulty comesfrom the fact that, from many points of view, transitivity isnot a really strong restriction on permutation groups, and thereare too many transitive groups [9, Sections 3 and 4]. In thispaper we solve the problem in the particular case of permutationgroups of prime power degree, proving the following result.1991 Mathematics Subject Classification 20B05, 20D60.     

On Transitive Permutation Groups with Primitive Subconstituents

Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05.     

Transitive Permutation Groups with Cyclic Point Stabilizers of Maximum Order

We prove that a transitive permutation group of degree n with a cyclic point stabilizer and whose order is n(n-1) is isomorphic to the affine group of degree 1 over a field with n elements. More generally we show that if a finite group G has an abelian and core-free Hall subgroup Q, then either Q has a small order (2|Q|² < |G|) or G is a direct product of 2-transitive Frobenius groups.     

Extension of Multiply Transitive Permutation Sets

Let G be a k-transitive permutation set on E and let E* = E∪{∞},∞ ? E; if G* is a (k: + 1)-transitive permutation set on E*, G* is said to be an extension of G whenever G _∝ ^* =G. In this work we deal with the problem of extending (sharply) k- transitive permutation sets into (sharply) (k + 1)-transitive permutation sets. In particular we give sufficient conditions for the extension of such sets; these conditions can be reduced to a unique one (which is a necessary condition too) whenever the considered set is a group. Furthermore we establish necessary and sufficient conditions for a sharply k- transitive permutation set (k ≥ 3) to be a group. Math. Subj. Class.: 20B20 Multiply finite transitive permutation groups 20B22 Multiply infinite transitive permutation groups     

Orbit-Homogeneity in Permutation Groups

This paper introduces the concept of orbit-homogeneity of permutationgroups: a group G is orbit-t-homogeneous if two sets of cardinalityt lie in the same orbit of G whenever their intersections witheach G-orbit have the same cardinality. For transitive groups,this coincides with the usual notion of t-homogeneity. Thisconcept is also compatible with the idea of partition transitivityintroduced by Martin and Sagan. Further, this paper shows that any group generated by orbit-t-homogeneoussubgroups is orbit-t-homogeneous, and that the condition becomesstronger as t increases up to [n/2], where n is the degree.So any group G has a unique maximal orbit-t-homogeneous subgroup_t(G), and _tG _t–1(G). Some structural results for orbit-t-homogeneousgroups, and a number of examples, are also given. 2000 MathematicsSubject Classification 20B10.     

Cycle-Closed Permutation Groups

A finite permutation group is cycle-closed if it contains all the cycles of all of its elements. It is shown by elementary means that the cycle-closed groups are precisely the direct products of symmetric groups and cyclic groups of prime order. Moreover, from any group, a cycle-closed group is reached in at most three steps, a step consisting of adding all cycles of all group elements. For infinite groups, there are several possible generalisations. Some analogues of the finite result are proved.     

Cofinitary Permutation Groups

A permutation group is cofinitary if any non-identity elementfixes only finitely many points. This paper presents a surveyof such groups. The paper has four parts. Sections 1–6develop some basic theory, concerning groups with finite orbits,topology, maximality, and normal subgroups. Sections 7–12give a variety of constructions, both direct and from geometry,combinatorial group theory, trees, and homogeneous relationalstructures. Sections 13–15 present some generalisationsof sharply k-transitive groups, including an orbit-countingresult with a character-theoretic flavour. The final sectiontreats some miscellaneous topics. Several open problems arementioned.     

On the Movement of Transitive Permutation p-Groups

Let G be a transitive permutation group on a set and m a positive integer. If | – | m for every subset of and all g G, then || 2mp/(p – 1) where p is the least odd prime dividing |G|. It was shown by Mann and Praeger [13] that, for p = 3, the 3-groups G which attain this bound have exponent p. In this paper we will show a generalization of this result for any odd primes.AMS Subject Classification (2000), 20BXX     

所有非平凡自同构均无固定点的有限群

设G是一个有限群,证明了G的每个非平凡自同构均为无固定点自同构当且仅当G是阿贝尔单群.作为应用,证明了群G的全形是以G为Frobenius核的Frobenius群当且仅当G是奇素数阶群.     

Permutation Groups in o-Minimal Structures

In this paper we develop a structure theory for transitive permutationgroups definable in o-minimal structures. We fix an o-minimalstructure M, a group G definable in M, and a set and a faithfultransitive action of G on definable in M, and talk of the permutationgroup (G, ). Often, we are concerned with definably primitivepermutation groups (G, ); this means that there is no propernon-trivial definable G-invariant equivalence relation on ,so definable primitivity is equivalent to a point stabiliserG being a maximal definable subgroup of G. Of course, sinceany group definable in an o-minimal structure has the descendingchain condition on definable subgroups [23] we expect many questionson definable transitive permutation groups to reduce to questionson definably primitive ones. Recall that a group G definable in an o-minimal structure issaid to be connected if there is no proper definable subgroupof finite index. In some places, if G is a group definable inM we must distinguish between definability in the full ambientstructure M and G-definability, which means definability inthe pure group G:= (G, .); for example, G is G-definably connectedmeans that G does not contain proper subgroups of finite indexwhich are definable in the group structure. By definable, wealways mean definability in M. In some situations, when thereis a field R definable in M, we say a set is R-semialgebraic,meaning that it is definable in (R, +, .). We call a permutationgroup (G, ) R-semialgebraic if G, and the action of G on canall be defined in the pure field structure of a real closedfield R. If R is clear from the context, we also just write‘semialgebraic’.     

Locally Finite Barely Transitive Groups

Infinite transitive permutation groups all proper subgroups of which have just finite orbits are treated. Under the extra condition of being locally finite, such groups are proved to be primary, and, moreover, soluble if the stabilizer of some point is soluble.     

Suborbits in Infinite Primitive Permutation Groups

For every infinite cardinal , we construct a primitive permutationgroup which has a finite suborbit paired with a suborbit ofsize . This answers a question of Peter M. Neumann. 2000 MathematicsSubject Classification 20B07, 20B15, 03C50, 05C20.     

Uncountable Cofinalities of Permutation Groups

A sufficient criterion is found for certain permutation groupsG to have uncountable strong cofinality, that is, they cannotbe expressed as the union of a countable, ascending chain (H_i)_i     

Permutation Groups and Covering Properties

The paper considers the possible cardinalities of maximal cofinitarygroups and the cofinalities of permutation groups on the setof natural numbers. These two cardinals are compared with someother well-known cardinal invariants which are related to coveringproperties in several forcing models. An application of combinatorialgroup theory to a construction of a forcing partially orderedset is presented.     

Codes from Affine Permutation Groups

It is shown that A(22,10) 50, A(23,10) 76, A(25,10) 166, A(26,10) 270, A(29,10) 1460, and A(28,12) 178, where A(n,d) denotes the maximum cardinality of a binary code of length n and minimum Hamming distance d. The constructed codes are invariant under permutations of some affine (or closely related) permutation group and have been found using computer search.     

Sharply k-Transitive Permutation Groups Viewed as Galois Groups

The classification of finite sharply k-transitive groups was achieved by the efforts of Jordan (1873), Dickson (1905), and Zassenhaus (1936). Likewise for other families of finite groups, one expects that they are realizable as Galois groups over the field of rational numbers \mathbbQ{\mathbb{Q}}. In this article, we study some properties of the polynomials f ? \mathbbQ[x]{f \in \mathbb{Q}[x]} such that the Galois group Gal(f) acts sharply k-transitively on its roots.