Among all groups $G$ of order $n$ which one will maximum the value: $\frac 1n\sum\_{g \in G}O(g)$ ? (Where $o(g)$ is the order of $g$). 



The Average of Orders 
As quid has stated the maximum attained for $\mathbb{Z}\_n$. The problem goes back to 1991. See Americam Mathematical Monthly 1991, page 970. 


