Normalizers of finite subgroups in homogenous symmetric groups and automorphisms

The structure of the centralizers of finite subgroups in homogenous symmetric groups S(χ(ξ)) are given up to isomorphism by Güven-Kegel-Kuzucuoğlu in 2015 and Kuzucuoğlu-Oliynyk-Sushchansky in 2018. In this article, we answer the natural question; what is the structure of the normalizers of finite semiregular subgroups in homogenous symmetric groups? We prove, NS(χ(ξ))(F)≅CS(χ(ξ))(F)⋊Aut(F)≅Σχ(ξ1)(F)⋊Aut(F) for some sequence ξ1. We answer negatively, the following open question: does every automorphism of S(χ(ξ)) preserve a level? We show that, there exist uncountably many automorphisms of S(χ(ξ)) which does not preserve any level. Finally, we show that the level preserving automorphisms of the homogenous finitary symmetric group FSym(κ)(χ(ξ)) is isomorphic to Sym(κ)×∏i∈NCSym(κ)(mi+1)(FSym(κ)(mi)).