Products of conjugate permutations

Discrete Mathematics, 1983

Using combinatorial methods, we will examine products of conjugacy classes in the symmetric group S, of all permutations of a countably infinite set. If p E S, has at least one infinite orbit in the underlying set and if s ES,, we give a characterization of when s is a product of two conjugates of p. From this, we derive that if four permutations pi E S, (i = 1,2,3,4) are given which all have infinite support, then any permutation of S, is a product of four elements conjugate to pi, p2, ps and p4, respectively. Similar results for permutations of uncountable sets are shown and classical groupaheoretical results are obtained from these theorems. Proof. We distinguish between two cases, p(l) = 0 and p(1) # 0. Case 1. p has no fixed points.

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.

Journal of Algebra, 1979

Groups St Andrews 2005

Until 1980, there was no such subgroup as 'infinite permutation groups', according to the Mathematics Subject Classification: permutation groups were assumed to be finite. There were a few papers, for example , and a set of lecture notes by Wielandt [72], from the 1950s. Now, however, there are far more papers on the topic than can possibly be summarised in an article like this one. I shall concentrate on a few topics, following the pattern of my conference lectures: the random graph (a case study); homogeneous relational structures (a powerful construction technique for interesting permutation groups); oligomorphic permutation groups (where the relations with other areas such as logic and combinatorics are clearest, and where a number of interesting enumerative questions arise); and the Urysohn space (another case study). I have preceded this with a short section introducing the language of permutation group theory, and I conclude with briefer accounts of a couple of topics that didn't make the cut for the lectures (maximal subgroups of the symmetric group, and Jordan groups). I have highlighted a few specific open problems in the text. It will be clear that there are many wide areas needing investigation!

Acta Mathematica Hungarica, 2012

We say that the subgroups G1 and G2 of a group G are mutually permutable if G1 permutes with every subgroup of G2 and G2 permutes with every subgroup of G1. Let G = G1G2. .. Gn be the product of its pairwise permutable subgroups G1, G2,. .. , Gn such that the product GiGj is mutually permutable. We investigate the structure of the finite group G if special properties of the factors G1, G2,. .. , Gn are known. Our results improve and extend some results of Asaad and Shaalan [1], Ezquerro and Soler-Escrivà [9] and Asaad and Monakhov [3].