Normalizers in homogeneous symmetric groups

Güven, Ülviye Büşra
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 $F$.  


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Berkman, A.; Kuzucuoğlu, Mahmut; OeZyurt, E. (2007-01-01)
We consider infinite locally finite-simple groups (that is, infinite groups in which every finite subset lies in a finite simple subgroup). We first prove that in such groups, centralizers of involutions either are soluble or involve an infinite simple group, and we conclude that in either case centralizers of involutions are not inert subgroups. We also show that in such groups, the centralizer of an involution is linear if and only if the ambient group is linear.
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Ercan, Gülin; Güloğlu, İsmail Ş. (Elsevier BV, 2008-7)
Let A be a finite abelian group that acts fixed point freely on a finite (solvable) group G. Assume that |G| is odd and A is of squarefree exponent coprime to 6. We show that the Fitting length of G is bounded by the length of the longest chain of subgroups of A.
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ERSOY, KIVANÇ; Kuzucuoğlu, Mahmut (2012-01-01)
Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite K-semisimple subgroups. Namely let G be a non-linear simple locally finite group which has a Kegel sequence K = {(G(i), 1) : i is an element of N} consisting of finite simple subgroups. Then for any finite subgroup F consisting of K-semisimple elements in G, the centralizer C-G(F) has an infinite abelian subgroup A is...
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ERSOY, KIVANÇ; Kuzucuoğlu, Mahmut; Shunwatsky, Pavel (2017-07-01)
Let p be a prime and G a locally finite group containing an elementary abelian p-subgroup A of rank at least 3 such that C-G(A) is Chernikov and C-G(a) involves no infinite simple groups for any a is an element of A(#). We show that G is almost locally soluble (Theorem 1.1). The key step in the proof is the following characterization of PSLp(k): An infinite simple locally finite group G admits an elementary abelian p-group of automorphisms A such that C-G(A) is Chernikov and C-G(A) Keywords: involves no inf...
Smooth manifolds with infinite fundamental group admitting no real projective structure
Çoban, Hatice; Ozan, Yıldıray; Department of Mathematics (2017)
In this thesis, we construct smooth manifolds with the infinite fundamental group Z_2*Z_2, for any dimension n>=4, admitting no real projective structure. They are first examples of manifolds in higher dimensions with infinite fundamental group admitting no real projective structures. The motivation of our study is the related work of Cooper and Goldman. They proved that RP^3#RP^3 does not admit any real projective structure and this is the first known example in dimension 3. 
Citation Formats
Ü. B. Güven, “Normalizers in homogeneous symmetric groups,” Ph.D. - Doctoral Program, Middle East Technical University, 2017.