Geodesic
Some criteria for the decomposability of finite groups
Matematičeskie zametki, Tome 12 (1972) no. 6, pp. 717-725.

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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. 
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     author = {V. M. Busarkin and N. D. Podufalov},
     title = {Some criteria for the decomposability of finite groups},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
     volume = {12},
     number = {6},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_12_6_a8/}
}
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V. M. Busarkin; N. D. Podufalov. Some criteria for the decomposability of finite groups. Matematičeskie zametki, Tome 12 (1972) no. 6, pp. 717-725. http://geodesic.mathdoc.fr/item/MZM_1972_12_6_a8/