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Vladikavkaz. Mat. Zh., 2018, Volume 20, Number 2, Pages 80–85 (Mi vmj656)  

On infinite Frobenius groups

D. V. Lytkinaab, V. D. Mazurovc, A. Kh. Zhurtovd

a Siberian State University of Telecommunications and Information Sciences
b Novosibirsk State University
c Sobolev Institute of Mathematics
d Kabardino-Balkar State University

Abstract: We study the structure of a periodic group $G$ satisfying the following conditions: $(F_1)$ The group $G$ is a semidirect product of a subgroup $F$ by a subgroup $H$; $(F_2)$ $H$ acts freely on $F$ with respect to conjugation in $G$, i. e. for $f\in F$, $h\in H$ the equality $f^h=f$ holds only for the cases $f=1$ or $h=1$. In other words $H$ acts on $F$ as the group of regular automorphisms. $(F_3)$ The order of every element $g\in G$ of the form $g=fh$ with $f\in F$ and $1\neq h\in H$ is equal to the order of $h$; in other words, every non-trivial element of $H$ induces with respect to conjugation in $G$ a splitting automorphism of the subgroup $F$. $(F_4)$ The subgroup $H$ is generated by elements of order $3$. In particular, we show that the rank of every principal factor of the group $G$ within $F$ is at most four. If $G$ is a finite Frobenius group, then the conditions $(F_1)$ and $(F_2)$ imply $(F_3)$. For infinite groups with $(F_1)$ and $(F_2)$ the condition $(F_3)$ may be false, and we say that a group is Frobenius if all three conditions $(F_1)$$(F_3)$ are satisfied. The main result of the paper gives a description of а periodic Frobenius groups with the property $(F_4)$.

Key words: periodic group, Frobenius group, free action, splitting automorphism.

Funding Agency Grant Number
Siberian Branch of Russian Academy of Sciences I.1.1., проект № 0314-2016-001


DOI: https://doi.org/10.23671/VNC.2018.2.14724

Full text: PDF file (238 kB)
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UDC: 512.54
Received: 19.01.2018

Citation: D. V. Lytkina, V. D. Mazurov, A. Kh. Zhurtov, “On infinite Frobenius groups”, Vladikavkaz. Mat. Zh., 20:2 (2018), 80–85

Citation in format AMSBIB
\Bibitem{LytMazZhu18}
\by D.~V.~Lytkina, V.~D.~Mazurov, A.~Kh.~Zhurtov
\paper On infinite Frobenius groups
\jour Vladikavkaz. Mat. Zh.
\yr 2018
\vol 20
\issue 2
\pages 80--85
\mathnet{http://mi.mathnet.ru/vmj656}
\crossref{https://doi.org/10.23671/VNC.2018.2.14724}
\elib{https://elibrary.ru/item.asp?id=35258720}


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