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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 $g$).

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For generalisation of this question and further information around it see also this MO question… Yet the situation regarding the generalistation seems a bit inconclusive. – user9072 Jan 3 '13 at 13:41
up vote 4 down vote accepted

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

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The maximum is attained for the cycylic group of order $n$, and only for this this group.

See Sums of element orders in finite groups Commincations in Algebra, Vol 37, 2009, which considers this problem (except for not dividing by $n$, which however changes nothing, since in the question the order of the group is fixed).

(The link is to the Zentralblatt MATH review and should work without subscription.)

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