Infinite Frobenius groups generated by elements of order 3. (English) Zbl 1477.20097

Yang, Nanying; Lytkina, Daria Victorovna; Mazurov, Victor Danilovich; Zhurtov, Archil Khazeshovich

Algebra Colloq. 27, No. 4, 741-748 (2020).

Let \(G\) be a group such that \(G=F\rtimes H\) where \(H\) acts fixed-point-freely on \(F\setminus \{1\}\) by conjugation, and \(|fh^{-1}|=|h|\) for each \(h\in H\setminus \{1\}\) of finite order and all \(f\in F\). Then \(G\) is called a Frobenius group.
Suppose that \(H\) is periodic and generated by elements of order \(3\). The main theorem states that \(H\) is then finite: specifically, \(H\cong C_3\) or \(\mathrm{SL}_2(3)\) or \(\mathrm{SL}_2(5)\). In the first case, \(F\) is nilpotent of class \(2\); otherwise, \(F\) is abelian. The theorem also describes the action of \(H\) on chief factors of \(G\) inside \(F\).

Reviewer: Dane Flannery (Galway)

MSC:

20H20	Other matrix groups over fields
20F50	Periodic groups; locally finite groups

Keywords:

Frobenius group; splitting automorphism; character table