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Let $G$ be a finite group and $nse\left(G\right)$ the set of numbers of elements with the same order in $G$. In this paper, we prove that a finite group $G$ is isomorphic to $M$, where $M$ is one of the Mathieu groups, if and only if the following hold: (1) $\left|G\right|=\left|M\right|$, (2) $nse\left(G\right)=nse\left(M\right)$.

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 $n\left(G\right)$ 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...