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Rank and Order of a Finite Group Admitting a Frobenius-Like Group of Automorphisms
Date
2014-07-01
Author
Ercan, Gülin
Khukhro, E. I.
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Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License
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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 above in terms of |H| and the rank of the fixed-point subgroup C (G) (H), and that |G| is bounded above in terms of |H| and |C (G) (H)|. As a corollary, in the case where G is an arbitrary finite group estimates are obtained of the form |G| a parts per thousand currency sign|C (G) (F)| center dot f(|H|, |C (G) (H)|) for the order, and r(G) a parts per thousand currency sign r(C (G) (F)) + g(|H|, r(C (G) (H))) for the rank, where f and g are some functions of two variables.
Subject Keywords
Order
,
Rank
,
Frobenius group
,
Finite group
,
Automorphism
URI
https://hdl.handle.net/11511/40525
Journal
ALGEBRA AND LOGIC
DOI
https://doi.org/10.1007/s10469-014-9287-4
Collections
Department of Mathematics, Article
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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...
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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...
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Ercan, Gülin (2019-03-01)
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.
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.
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G. Ercan and E. I. Khukhro, “Rank and Order of a Finite Group Admitting a Frobenius-Like Group of Automorphisms,”
ALGEBRA AND LOGIC
, pp. 258–265, 2014, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/40525.