ON A QUESTION OF JAIKIN-ZAPIRAIN ABOUT THE AVERAGE ORDER ELEMENTS OF FINITE GROUPS

For a finite group G, the average order o(G) is defined to be the average of all order elements in G, that is o(G) = 1 |G| P x∈G o(x), where o(x) is the order of element x in G. Jaikin- Zapirain in [On the number of conjugacy classes of finite nilpotent groups, Advances in Mathematics, 227 (2011) 1129-1143] asked the following question: if G is a finite (p-) group and N is a normal (abelian) subgroup of G, is it true that o(N) 1 2 ≤ o(G)? We say that G satisfies the average condition if o(H) ≤ o(G), for all subgroups H of G. In this paer we show that every finite abelian group satisfies the average condition. This result confirms and improves the question of Jaikin-Zapirain for finite abelian groups.