Approximate centralizers in the group of homeomorphisms of the rational numbers
This note is concerned with the structure of certain subgroups G (plural ``subgroups'' because there is one for each prime number p) of the symmetric group S consisting of all permutations of \({\mathbb{Z}}\). Let \(h\in S\) be the translation \(x\mapsto x+1\), let p be a prime number and let \(C_ n\) be the centralizer in S of \(h^{p^ n}\), so that \(<h>=C_ 0<C_ 1<C_ 2... \). Then G is defined to be \(\cup C_ n\). These groups were called ``approximate centralizers'' by the reviewer [J. Lond. Math. Soc., II. Ser. 32, 439-448 (1985; Zbl 0587.20001)]; a similar group has been studied more recently by \textit{G. Higman} [J. Algebra 131, No.1, 359- 369 (1990)]. The main result of the present note is that the commutator subgroup \(G'\) is simple. Furthermore, if \(p=2\) then \(G/G'\cong {\mathbb{Z}}\) and if p is odd then \(G/G'\cong {\mathbb{Z}}\times {\mathbb{Z}}_ 2\). The proofs are based upon the fact that \(C_ n\) is a wreath product \({\mathbb{Z}} Wr Sym(p^ n)\) and consist of analysis of the embedding of \(C_ n\) into \(C_{n+1}\).
scientific article; zbMATH DE number 4168924