Geodesic
Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 827-831.

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Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups $G$ whose set of numbers of subgroups of possible orders $n(G)$ has exactly two elements. We show that if $G$ is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then $G$ has a normal Sylow subgroup of prime order and $G $ is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with $n(G)=\{1, q+1\}$ for some prime $q$. In particular, $G$ is supersolvable under this condition.
DOI : 10.1007/s10587-014-0135-4
Classification : 20E07, 20E45
Keywords: finite group; number of subgroups of possible orders 
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     title = {Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements},
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Shao, Changguo; Jiang, Qinhui. Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 827-831. doi : 10.1007/s10587-014-0135-4. http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0135-4/

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