Frobenius groups of automorphisms with almost fixed point free kernel

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2019-03-01

Author

Ercan, Gülin

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Let FH be a Frobenius group with kernel F and complement H, acting coprimely on the finite solvable group G by automorphisms. We prove that if C-G(H) is of Fitting length n then the index of the n-th Fitting subgroup F-n(G) in G is bounded in terms of vertical bar C-G(F)vertical bar and vertical bar F vertical bar. This generalizes a result of Khukhro and Makarenko [6] which handles the case n = 1.

Subject Keywords

Solvable group, Automorphism, Fitting length, Frobenius group

URI

https://hdl.handle.net/11511/37246

Journal

JOURNAL OF ALGEBRA

DOI

https://doi.org/10.1016/j.jalgebra.2018.12.004

Collections

Department of Mathematics, Article

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NILPOTENT LENGTH OF A FINITE SOLVABLE GROUP WITH A FROBENIUS GROUP OF AUTOMORPHISMS Ercan, Gülin; Ogut, Elif (2014-01-01) We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C-CG(F) (h) = 1 for all nonidentity elements h is an element of H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.
Frobenius-like groups as groups of automorphisms Ercan, Gülin; Khukhro, Evgeny (2014-01-01) 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 C-G(H) and focusing mainly on bounds for the Fitting height and rela...
Rank and Order of a Finite Group Admitting a Frobenius-Like Group of Automorphisms Ercan, Gülin; Khukhro, E. I. (2014-07-01) 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]. Suppose that a finite group G admits a Frobenius-like group of automorphisms FH of coprime order with certain additional restrictions (which are satisfied, in particular, if either |FH| is odd or |H| = 2). In the case where G is a finite p-group such that G = [G, F] it is proved that the rank of G is bounded ab...
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.
Derived length of a Frobenius-like kernel Ercan, Gülin; Khukhro, Evgeny (2014-08-15) A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F called kernel which has a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]. Suppose that a Frobenius-like group FH acts faithfully by linear transformations on a vector space V over a field of characteristic that does not divide vertical bar FH vertical bar. It is proved that the derived length of the kernel F is bounded solely in terms of the dimension m = dim C-V(H...

Citation Formats

G. Ercan, “Frobenius groups of automorphisms with almost fixed point free kernel,” JOURNAL OF ALGEBRA, pp. 384–389, 2019, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/37246.