On infinite Frobenius groups

D. V. Lytkina ; V. D. Mazurov ; A. Kh. Zhurtov

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 2, pp. 80-85

Résumé

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 $(F_4)$.