Gülin ERCAN, Ismail Suayip GÜLOĞLU, Evgeny KHUKHRO

Frobenius-like groups as groups of automorphisms

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]. Such subgroups and sections are abundant in any nonnilpotent finite group. We discuss several recent results about the properties of a finite group G admitting a Frobenius-like group of automorphisms FH aiming at restrictions on G in terms of CG(H) and focusing mainly on bounds for the Fitting height and related parameters. Earlier such results were obtained for Frobenius groups of automorphisms; new theorems for Frobenius-like groups are based on new representation-theoretic results. Apart from a brief survey, the paper contains the new theorem on almost nilpotency of a finite group admitting a Frobenius-like group of automorphisms with fixed-point-free almost extraspecial kernel.

Anahtar Kelimeler:

Frobenius group, Frobenius-like group, fixed points, Fitting height, nilpotency class, derived length, rank, order

Frobenius-like groups as groups of automorphisms

Keywords:

Frobenius group, Frobenius-like group, fixed points, Fitting height, nilpotency class, derived length, rank, order,

PDF

___

G¨ulo˘glu ˙IS¸, Ercan G. Action of a Frobenius-like group. J Algebra 2014; 402: 533–543.
Ercan G, G¨ulo˘glu ˙IS¸. Action of a Frobenius-like group with fixed-point-free kernel. J Group Theory doi: 10.1515/jgt- 2014-0002.
Ercan G, G¨ulo˘glu ˙IS¸, Khukhro EI. Rank and Order of a Finite Group admitting a Frobenius-like Group of Automorphisms. submitted to Algebra and Logic, 2014.
Ercan G, G¨ulo˘glu ˙IS¸, Khukhro EI. Derived length of a Frobenius-like kernel. J Algebra http://dx.doi.org/10.1016/j.jalgebra.2014.04.0252014.
Ercan G, G¨ulo˘glu ˙IS¸, ¨O˘g¨ut E. Nilpotent length of a Finite Solvable Group with a coprime Frobenius Group of Automorphisms. Communications in Algebra 2014; 42: 4751–4756.
Belyaev VV, Hartley B. Centralizers of finite nilpotent subgroups in locally finite groups. Algebra Logika 1996; 35: 389–410; English transl. Algebra Logic 1996; 35: 217–228.
Huppert B. Endliche Gruppen I. Springer-Verlag, Berlin-New York, 1967.
Isaacs IM. Finite Group Theory. Amer Math Soc, Providence: Graduate Studies in Mathematics 2008; 92.
Khukhro EI. The nilpotent length of a finite group admitting a Frobenius group of automorphisms with a fixed- point-free kernel. Algebra Logika 2010; 49: 819–833; English transl, Algebra Logic 2011; 49: 551–560.
Khukhro EI. Fitting height of a finite group with a Frobenius group of automorphisms. J Algebra 2012; 366: 1–11.
Khukhro EI. Rank and order of a finite group admitting a Frobenius group of automorphisms. Algebra Logika 2013; 52: 99–108; English transl., Algebra Logic 2013; 52: 72–78.
Khukhro EI, Makarenko NY. Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms. J. Algebra 2013; 386: 77–104.
Khukhro EI, Makarenko NY. Finite p -groups admitting a Frobenius groups of automorphisms with kernel a cyclic p -group, to appear in Proc Amer Math Soc 2014.
Khukhro EI, Makarenko NY, Shumyatsky P. Frobenius groups of automorphisms and their fixed points. Forum Math. 2014; 26: 73–112.
Makarenko NY, Shumyatsky P. Frobenius groups as groups of automorphisms. Proc Amer Math Soc 2010; 138: 3425–3436.
Shumyatsky P. On the exponent of a finite group with an automorphism group of order twelve. J Algebra 2011; 331: 482–489.
Unsolved problems in group theory. The Kourovka notebook. 2010; 17th ed. Institute of Mathematics. Novosibirsk, 20