tailieunhanh - On finitary permutation groups

In this work we give some sufficient conditions under which the structure of a transitive group of finitary permutations on an infinite set can be determined from the structure of a point stabilizer. | Turk J Math 30 (2006) , 101 – 116. ¨ ITAK ˙ c TUB On Finitary Permutation Groups Ali Osman Asar Abstract In this work we give some sufficient conditions under which the structure of a transitive group of finitary permutations on an infinite set can be determined from the structure of a point stabilizer. Also, we give some sufficient conditions for the existence of a proper subgroup having an infinite orbit in a totally imprimitive pgroup of finitary permutations. These results, with the help of some known results, give sufficient conditions for the nonexistence of a perfect locally finite minimal non F C - (p-group). Key Words: Finitary permutation, primitive, almost primitive, totally imprimitive. 1. Introduction Let G be a transitive subgroup of F Sym(Ω), where Ω is infinite. Many authors have investigated G by imposing suitable conditions on a point stabilizer (see, for example,[1], [2], [4], [7], [9]). Also another problem which might be interesting is finding sufficient conditions under which G can have a proper subgroup having an infinite orbit. In view of the important reduction theorems given in [3] and [8], any solution of the last problem means a solution for the following well known problem: Does there exist a perfect minimal non F C - (p-group)? The aim of this work is to obtain some sufficient conditions about the problems described above. Let Ω be a (possibly infinite)set and let Sym(Ω) be the symmetric group on Ω. For each x ∈ Sym(Ω) the set supp(x) = {i ∈ Ω : x(i) 6= i} is called the support of x and if supp(x) is finite, then x is called a finitary permutation. The set of all the finitary permutations on Ω forms a subgroup which is denoted by F Sym(Ω). Let G be a subgroup 2000 AMS Mathematics Subject Classification: 20B 07 20B 35 20E 25 101 ASAR of Sym(Ω) and a ∈ Ω. Then Ga = {g ∈ G : g(a) = a} is called the stabilizer of a in G and G(a) = {g(a) : a ∈ G} is called the orbit of G containing a. More generally if ∆ is a nonempty subset of Ω,