Classes of universal words for the infinite symmetric groups (Q1061218)

In the literature several authors have dealt with the problem of characterizing for various families Z of groups the set of all Z- universal words. The paper is devoted to this problem for the family ISym of all infinite symmetric groups \(S_{\nu}\) (\(\nu\geq 0)\), where \(S_{\nu}\) denotes the group of all permutations of a set of cardinality \(\aleph_{\nu}\). First, the generalization of earlier results is given. Next a new class of arbitrarily complex ISym-universal words is presented. The paper ends with a characterization of those simple factor- groups H in the Jordan-Hölder decomposition series of \(S_{\nu}\) which are verbally complete, i.e. in which any non-trivial word is H-universal.