Locally finite groups and their subgroups with small centralizers

Kuzucuoğlu, Mahmut
Shunwatsky, Pavel
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 infinite simple groups for any a is an element of A(#) if and only Locally finite group if G is isomorphic to PSLp(k) for some locally finite field k Centralizer of characteristic different from p and A has order p(2). (C) 2017 Published by Elsevier Inc.


On local finiteness of periodic residually finite groups
Kuzucouoglu, M; Shumyatsky, P (2002-10-01)
Let G be a periodic residually finite group containing a nilpotent subgroup A such that C-G (A) is finite. We show that if [A, A(g)] is finite for any g is an element of G, then G is locally finite.
Inert subgroups and centralizers of involutions in locally finite simple groups
Özyurt, Erdal; Kuzucuoğlu, Mahmut; Department of Mathematics (2003)
A subgroup H of a group G is called inert if [H : H \ Hg] is finite for all g 2 G. A group is called totally inert if every subgroup is inert. Among the basic properties of inert subgroups, we prove the following. Let M be a maximal subgroup of a locally finite group G. If M is inert and abelian, then G is soluble with derived length at most 3. In particular, the given properties impose a strong restriction on the derived length of G. We also prove that, if the centralizer of every involution is inert in an...
Pamuk, Semra (2014-07-03)
Let G be a finite group and F be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative F-projective resolution for Z when F is the family of all subgroups HG with rkHrkG-1. We answer this question negatively by calculating the relative group cohomology FH*(G, ?(2)) where G = Z/2xZ/2 and F is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology FH*(G, M) can be calc...
Prime graphs of solvable groups
Ulvi , Muhammed İkbal; Ercan, Gülin; Department of Electrical and Electronics Engineering (2020-8)
If $G$ is a finite group, its prime graph $Gamma_G$ is constructed as follows: the vertices are the primes dividing the order of $G$, two vertices $p$ and $q$ are joined by an edge if and only if $G$ contains an element of order $pq$. This thesis is mainly a survey that gives some important results on the prime graphs of solvable groups by presenting their proofs in full detail.
Existentially closed groups
Gürel, Yağmur; Kuzucuoğlu, Mahmut; Koçak Benli, Dilber; Department of Mathematics (2021-12-10)
A group G is called an Existentially Closed Group (Algebraically Closed Group) if for every finite system of equations and inequations with coefficients from G which has a solution in an over group H ≥ G, has a solution in G. Existentially closed groups were introduced by W. R. Scott in 1951. The notion of existentially closed groups is close to the notion of algebraically closed fields but there are substantial differences. Existentially closed groups were studied and advanced by B. H. Neumann. In this sur...
Citation Formats
K. ERSOY, M. Kuzucuoğlu, and P. Shunwatsky, “Locally finite groups and their subgroups with small centralizers,” JOURNAL OF ALGEBRA, pp. 1–11, 2017, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/48896.