On Infinite Frobenius Groups

Mazurov V. D. , Zhurtov A. H. , Lytkina D. V.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 2.

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 à periodic Frobenius groups with the property \((F_4)\).

Keywords: periodic group, Frobenius group, free action, splitting automorphism

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