Two Criteria for Nonsimplicity of a Group Possessing a Strongly Embedded Subgroup and a Finite Involution

A proper subgroup $H$ of a group $G$ is said to be strongly embedded if $2\in\pi (H)$ and $2\notin\pi(H\cap H^g)$ ($\forall g\in G\setminus H$). An involution $i$ of $G$ is said to be finite if $|ii^g|<\infty$ ($\forall g\in G$). As is known, the structure of a (locally) finite group possessing a strongly embedded subgroup is determined by the theorems of Burnside and Brauer–Suzuki, provided that the Sylow 2-subgroup contains a unique involution. In this paper, sufficient conditions for the equality $m_2(G)=1$ are established, and two analogs of the Burnside and Brauer–Suzuki theorems for infinite groups $G$ possessing a strongly embedded subgroup and a finite involution are given.