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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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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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.

Citation Formats

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.