Abstract:
Suppose that a finite simple group $G$ containing a strongly isolated subgroup whose order is divisible by $3$ has a $2$-local subgroup whose order is also divisible by $3$. Then $G$ is isomorphic either to $\operatorname{PSL}(3,4)$ or to $\operatorname{PSL}(2,q)$ for
a suitable $q$.
If a finite simple group $G$ contains for some prime number $p\in\{3,5\}\cap\pi(G)$ a strongly isolated subgroup whose order is divisible by $p$, then $G$ is isomorphic to one of the groups $\operatorname{PSL}(3,4)$, $\operatorname{PSL}(2,q)$ for a suitable $q$,
or $\operatorname{Sz}(2^{2m+1})$, $m>0$.
A number of other results on groups containing strongly isolated subgroups are also derived in the paper.
Bibliography: 13 titles.
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