Geodesic
Product of a biprimary and a 2-decomposable group
Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 641-649 
V. S. Monakhov. Product of a biprimary and a 2-decomposable group. Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 641-649. http://geodesic.mathdoc.fr/item/MZM_1978_23_5_a0/
@article{MZM_1978_23_5_a0,
     author = {V. S. Monakhov},
     title = {Product of a biprimary and a 2-decomposable group},
     journal = {Matemati\v{c}eskie zametki},
     pages = {641--649},
     year = {1978},
     volume = {23},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_5_a0/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Suppose a finite group $G$ is the product of a subgroups $A$ and $B$ of coprime orders, and suppose the order of $A$ is $p^aq^b$, where $p$ and $q$ are primes, and $B$ is a 2-decomposable group of even order. Assume that a Sylow $p$-subgroup $P$ is cyclic. If the order of $P$ is not equal to 3 or 7, then $G$ is solvable. If $G$ is nonsolvable and $G$ contains no nonidentity solvable invariant subgroups, then $G$ is isomorphic to $PSL(2, 7)$ or $PGL(2, 7)$.