Geodesic
The product of two groups, one of which contains a~cyclic subgroup of index $\le2$
Matematičeskie zametki, Tome 16 (1974) no. 2, pp. 285-295

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We prove that a finite group $G=A\cdot B$ is solvable if the groups $A$ and $B$ contain cyclic subgroups with indices $\le2$. We provide a description of two classes of nonsolvable factorizable groups. 
@article{MZM_1974_16_2_a12,
     author = {V. S. Monakhov},
     title = {The product of two groups, one of which contains a~cyclic subgroup of index $\le2$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {285--295},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_2_a12/}
}
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V. S. Monakhov. The product of two groups, one of which contains a~cyclic subgroup of index $\le2$. Matematičeskie zametki, Tome 16 (1974) no. 2, pp. 285-295. http://geodesic.mathdoc.fr/item/MZM_1974_16_2_a12/