Normalizers of finite subgroups in homogenous symmetric groups and automorphisms

Guven U. B., Kuzucuoğlu M.

COMMUNICATIONS IN ALGEBRA, vol.47, pp.5182-5191, 2019 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 47
  • Publication Date: 2019
  • Doi Number: 10.1080/00927872.2019.1612423
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.5182-5191
  • Keywords: Centralizer, diagonal embedding, level-preserving automorphisms, normalizer
  • Middle East Technical University Affiliated: Yes


The structure of the centralizers of finite subgroups in homogenous symmetric groups S(chi(xi)) are given up to isomorphism by Guven-Kegel-Kuzucuoglu in 2015 and Kuzucuoglu-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, N-S(chi(xi))(F) congruent to C-S(chi(xi) (F) x Aut(F) congruent to Sigma(chi(xi 1))(F) x Aut(F) for some sequence xi(1): We answer negatively, the following open question: does every automorphism of S(chi(xi)) preserve a level? We show that, there exist uncountably many automorphisms of S(chi(xi)) which does not preserve any level. Finally, we show that the level preserving automorphisms of the homogenous finitary symmetric group FSym(kappa)(chi(xi)) is isomorphic to Sym(kappa) x Pi(i is an element of N) CSym(kappa)(mi+1) (FSym(kappa)(m(i))).