Some criteria for the decomposability of finite groups

In this paper we prove the following fundamental results. \underline{Theorem 1}: A finite unsolvable group, every involution of which is contained in a proper isolated subgroup, is decomposable. \underline{Theorem 2}: Suppose the finite unsolvable group $G$ contains a strongly isolated subgroup $M$ of odd order with isolated normalizer $N(M)$ of even order. If $|N(M):(M)|>2$, the group $G$ is isomorphic with one of the groups: 1) $PSL(2,q)$, $q$ odd; 2) $PGL(2,q)$, $q$ odd.