On groups with a strongly embedded unitary subgroup

Abstract

The proper subgroup $B$ of the group $G$ is called {\it strongly embedded}, if $2\in\pi(B)$ and $2\notin\pi(B \cap B^g)$ for any element $g \in G \setminus B $ and, therefore, $ N_G(X) \leq B$ for any 2-subgroup $ X \leq B $. An element $a$ of a group $G$ is called {\it finite} if for all $ g\in G $ the subgroups $ \langle a, a^g \rangle $ are finite. In the paper, it is proved that the group with finite element of order $4$ and strongly embedded subgroup isomorphic to the Borel subgroup of $U_3(Q)$ over a locally finite field $Q$ of characteristic $2$ is locally finite and isomorphic to the group $U_3(Q)$. Keywords: A strongly embedded subgroup of a unitary type, subgroups of Borel, Cartan, involution, finite element.