Universal for Infinite Symmetric Groups ?
نویسنده
چکیده
Let W= W(xx,...,Xj)bc any word in thej free generators xx,...,Xj, and suppose that W cannot be expressed in the form W = Vk for V a word and for k an integer with \k\¥" 1. We ask whether the equation f=W has a solution (xx>... ,Xj) = (a,,... ,Oj) €E GJ whenever G is an infinite symmetric group and/is an element in G. We establish an affirmative answer in the case that W(x, y) = xmy" for m and n nonzero integers.
منابع مشابه
Remarks on Actions on Compacta by Some Infinite-dimensional Groups
We discuss some techniques related to equivariant compactifications of uniform spaces and amenability of topological groups. In particular, we give a new proof of a recent result by Glasner and Weiss describing the universal minimal flow of the infinite symmetric group S∞ with the standard Polish topology, and extend Bekka’s concept of an amenable representation, enabling one to deduce non-amen...
متن کاملInfinite-dimensional General Linear Groups Are Groups of Universally Finite Width
Recently George Bergman proved that the symmetric group of an infinite set possesses the following property which we call by the universality of finite width: given any generating set X of the symmetric group of an infinite set Ω, there is a uniform bound k such that any permutation σ ∈ Sym(Ω) is a product of at most k elements of X∪X. Bergman also formulated a sort of general conjecture statin...
متن کاملUniversal Abelian Groups
We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For n ≥ 2, there is a purely universal separable p-group in אn if, and only if, 20 ≤ אn. §0 Introduction In this paper “group” will always mean “infinite abelian group”, and “cardina...
متن کاملBi- Logic Via Infinite Singletons
Bi-logic [1] describes two sides of the human thinking, the rational reasoning (asymmetric mode) and the symmetric mode, also termed indivisible mode, where any relation is symmetric and any set is infinite. In a quantum model [2], we have characterized the class of finite sets for which the membership relation can be expressed as a finite propositional disjunction of equalities, in the object ...
متن کاملSubgroups of Infinite Symmetric Groups
This paper and its sequel [17] deal with a range of questions about the subgroup structure of infinite symmetric groups. Our concern is with such questions as the following. How can an infinite symmetric group be expressed as the union of a chain of proper subgroups? What are the subgroups that supplement the normal subgroups of an infinite symmetric group? What are the maximal proper subgroups...
متن کامل