On a~generalization of Frobenius' theorem to infinite groups

A. I. Sozutov ; V. P. Shunkov

Sbornik. Mathematics, Tome 29 (1976) no. 4, pp. 441-451

Résumé

In this paper the following theorem is proved. Theorem. Suppose $G$ is a group, $H$ is a subgroup, and $a$ is an element of prime order $p\ne2$ in $H$ such that a) {\it$(G, H)$ is a Frobenius pair, i.e. $H\cap g^{-1}Hg=1$ for all $g\in G\setminus H$}; b) {\it for any $g\in G\setminus H$ the group $\langle a,g^{-1}ag\rangle$ is finite. Then $G = F_p\leftthreetimes H$, where $F_p$ is a periodic group containing no $p$-elements, and either $H$ possesses a unique involution or $H=N_G(\langle a\rangle)$.} Examples of periodic groups are given to show that the conditions $p\ne2$ and b) are essential restrictions in the theorem. It is proved that in the class of periodic biprimitively finite groups the existence in a group $G$ of a Frobenius pair $(G, H)$ already implies that $G=F_p\leftthreetimes H$ and $G$ admits a partition, i.e. $F^\#_p = F_p\setminus\{1\}=G\setminus\bigcup_{x\in G}H^x$. Bibliography: 14 titles.