On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period

A proper subgroup $H$ of a group $G$ is said to be strongly isolated if it contains the centralizer of any nonidentity element of $H$ and 2-isolated if the conditions $C_G(g)\cap H\ne1$ and $2\in\pi(C_G(g))$ imply that $C_G(g)\le H$. An involution $i$ in a group $G$ is said to be finite if $|ii^g|<\infty$ ($\forall g\in G$). In the paper we study a group $G$ with finite involution $i$ and with a 2-isolated locally finite subgroup $H$ containing an involution. It is proved that at least one of the following assertions holds:

• 1) all 2-elements of the group $G$ belong to $H$;
• 2) $(G,H)$ is a Frobenius pair, $H$ coincides with the centralizer of the only involution in $H$, and all involutions in $G$ are conjugate;
• 3) $G=F\leftthreetimes C_G(i)$ is a locally finite Frobenius group with Abelian kernel $F$;
• 4) $H=V\leftthreetimes D$ is a Frobenius group with locally cyclic noninvariant factor $D$ and a strongly isolated kernel $V$, $U=O_2(V)$ is a Sylow 2-subgroup of the group $G$, and $G$ is a $Z$-group of permutations of the set $\Omega=\{U^g\mid g\in G\}$.