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.