Centralizers of involutions in locally finite-simple groups

Berkman, A.
Kuzucuoğlu, Mahmut
OeZyurt, E.
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.


Centralizers of subgroups in simple locally finite groups
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...
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.
Fixed point free action on groups of odd order
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.
Centralizers of abelian subgroups in locally finite simple groups
Kuzucuoğlu, Mahmut (1997-06-01)
It is shown that, if a non-linear locally finite simple group is a union of finite simple groups, then the centralizer of every element of odd order has a series of finite length with factors which are either locally solvable or non-abelian simple. Moreover, at least one of the factors is non-linear simple. This is also extended to abelian subgroup of odd orders.
Centralizers of finite subgroups in simple locally finite groups
Ersoy, Kıvanç; Kuzucuoğlu, Mahmut; Department of Mathematics (2009)
A group G is called locally finite if every finitely generated subgroup of G is finite. In this thesis we study the centralizers of subgroups in simple locally finite groups. Hartley proved that in a linear simple locally finite group, the fixed point of every semisimple automorphism contains infinitely many elements of distinct prime orders. In the first part of this thesis, centralizers of finite abelian subgroups of linear simple locally finite groups are studied and the following result is proved: If G ...
Citation Formats
A. Berkman, M. Kuzucuoğlu, and E. OeZyurt, “Centralizers of involutions in locally finite-simple groups,” RENDICONTI DEL SEMINARIO MATEMATICO DELLA UNIVERSITA DI PADOVA, pp. 189–196, 2007, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/53360.