Fixed point free action on groups of odd order

Ercan, Gülin
Güloğlu, İsmail Ş.
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.
Journal of Algebra


Fixed-point free action of an abelian group of odd non-squarefree exponent
Ercan, Gülin; Güloğlu, İsmail Ş.; Sağdiçoğlu, Öznur Mut (Cambridge University Press (CUP), 2011-1-19)
<jats:title>Abstract</jats:title><jats:p>Let <jats:italic>A</jats:italic> be a finite group acting fixed-point freely on a finite (solvable) group <jats:italic>G</jats:italic>. A longstanding conjecture is that if (|<jats:italic>G</jats:italic>|, |<jats:italic>A</jats:italic>|) = 1, then the Fitting length of <jats:italic>G</jats:italic> is bounded by the length of the longest chain of subgroups of <jats:italic>A</jats:italic>. It is expected that the conjecture is true when the coprimeness condition is rep...
Normalizers in homogeneous symmetric groups
Güven, Ülviye Büşra; Kuzucuoğlu, Mahmut; Department of Mathematics (2017)
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 embedding...
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.
Centralizers of involutions in locally finite-simple groups
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.
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...
Citation Formats
G. Ercan and İ. Ş. Güloğlu, “Fixed point free action on groups of odd order,” Journal of Algebra, pp. 426–436, 2008, Accessed: 00, 2020. [Online]. Available: