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Frobenius groups of automorphisms with almost fixed point free kernel
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Date
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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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.
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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.
Beauville structures in p-groups
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Given a finite group G and two elements x, y in G, we denote by Sigma(x,y) the union of all conjugates of the cyclic subgroups generated by x, y and xy. Then G is called a Beauville group of unmixed type if the following conditions hold: (i) G is a 2-generator group. (ii) G has two generating sets {x1,y1} and {x2, y2} such that Sigma (x1, y1) intersection Sigma(x2, y2) is 1. In this case, {x1, y1} and {x2, y2} are said to form a Beauville structure for G. The main purpose of this thesis is to extend the kn...
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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.