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Lytkina, Dar’ya Viktorovna; Mazurov, Viktor Danilovich; Zhurtov, Archil Khazeshovich

On infinite Frobenius groups. (Russian. English summary) Zbl 1445.20034

Vladikavkaz. Mat. Zh. 20, No. 2, 80-85 (2018).
Summary: 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 a periodic Frobenius groups with the property \((F_4)\).

MSC:

20F50 Periodic groups; locally finite groups
20E34 General structure theorems for groups
20F28 Automorphism groups of groups

Keywords:

periodic group; Frobenius group; free action; splitting automorphism

Software:

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\textit{D. V. Lytkina} et al., Vladikavkaz. Mat. Zh. 20, No. 2, 80--85 (2018; Zbl 1445.20034)
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References:

[1] Mazurov V. D., “A generalization of a theorem of Zassenhaus”, Vladikavkaz Math. J., 10:1 (2008), 40-52 (in Russian) · Zbl 1324.20007
[2] Zhurtov A. Kh., “On regular automorphisms of order 3 and frobenius pairs”, Siberian Math. J., 41:2 (2000), 268-275 · Zbl 0956.20036
[3] GAP: Groups, algorithms, and programming,
[4] Isaacs I. M., Character theory of finite groups, American Math. Soc. Chelsea Publ., Providence (R. I.), 2006, 304 pp. · Zbl 1119.20005
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