Normalizers in homogeneous symmetric groups

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2017

Öğrenci: ÜLVİYE BÜŞRA GÜVEN

Danışman: MAHMUT KUZUCUOĞLU

Özet:

We study some properties of locally finite simple groups, which are the direct limit of finite (finitary) symmetric groups of (strictly) diagonal type. The direct limit of the finite (finitary) symmetric groups of strictly diagonal type is called textbf{homogeneous (finitary) symmetric groups}. In cite{gkk}, Kegel, Kuzucuou{g}lu and myself studied the structure of centralizer of finite groups in the homogeneous finitary symmetric groups. Instead of strictly diagonal embeddings, if we have diagonal embeddings, we will have direct limit of finite symmetric groups of diagonal type. We prove the centralizer of a finite subgroup for the symmetric groups of diagonal type is the direct product of homogeneous monomial groups and a symmetric group of diagonal type. We also study the level preserving automorphisms of the symmetric groups of diagonal type and finitary homogeneous symmetric groups. We prove that the level preserving automorphisms of both groups is isomorphic to the Cartesian product of centralizers of subgroups. In the last part of the thesis, we study the normalizers of finite subgroups in both homogeneous symmetric groups and homogeneous finitary symmetric groups. In the first class, we find normalizers of finite semi-regular subgroups and in the latter class we find normalizers of finite subgroups, $F$, satisfying $F_alpha= F mbox{or} 1$. In each class of groups, the quotient of the normalizer of finite subgroup, $F$, with the centralizer is isomorphic to the automorphism group of $F$.