On a generalization of Frobenius' theorem to infinite groups

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. $F^\#_p = F_p\setminus\{1\}=G\setminus\bigcup_{x\in G}H^x$.
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