Let $G$ be a finite group of order $n$ and denote by $\pi\_e(G)$ the set of element orders of $G$. What can be said about $G$ if $\pi\_e(G)$ forms a sublattice of the lattice of divisors of $n$? 



A question on the set of element orders of a finite group
Let $G$ be a finite group, $n(G)$ the l.c.m. of orders of elements in $G$. Here are some obvious observations. A group $G$ belongs to your class $\mathcal C$ iff $G$ contains the cyclic group of order $n(G)$. Every $p$-group belongs to $\mathcal C$. The class is closed under direct products of groups with co-prime orders. Hence it contains all finite nilpotent groups. Hence the class is closed under all (finite) direct products. For every group $G$, the group $G\times {\mathbb Z}/n(G){\mathbb Z}$ is in $\mathcal C$. 


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