Geodesic
$ABA$-groups with cyclic subgroup~$B$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 3, pp. 10-22

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Some criteria to the solubility of groups of the form $G=ABA$ with a nilpotent subgroup $A$ and a cyclic subgroup $B$ are derived. In particular, it is proved (using the classification of the finite simple groups) that the finite group $G=ABA$ is soluble if $A$ is a nilpotent group of odd order and $B$ is a cyclic group and $(|A|,|B|)=1$.
Mots-clés : simple group, Lie type group, sporadic simple group. 
@article{TIMM_2012_18_3_a1,
     author = {B. Amberg and L. S. Kazarin},
     title = {$ABA$-groups with cyclic subgroup~$B$},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {10--22},
     publisher = {mathdoc},
     volume = {18},
     number = {3},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2012_18_3_a1/}
}
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B. Amberg; L. S. Kazarin. $ABA$-groups with cyclic subgroup~$B$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 3, pp. 10-22. http://geodesic.mathdoc.fr/item/TIMM_2012_18_3_a1/