Let G be a finite group. The average order o(G) is defined to be the average of all order elements in G. We say that G satisfies the average condition if o(H)< o(G) for every subgroup H of G. Our main purpose is to study the structure of finite groups using their average orders. We show that every finite abelian group satisfies the average condition, which confirms and improves the question of Jaikin-Zapirain for abelian groups. Also, we classify minimal non-abelian groups which satisfy the average condition. We continue the study of finite groups in terms of the number of average order of its non-isomorphic subgroups. We define the set X(G) be the set of the average order of non-isomorphic subgroups of G. We prove that |X(G)|<3 if and only if G is isomorphic to 1, C_p, C^2_p or C_p^2. We show that if G is a finite group, then |X(G)|=4 if and only if either G has order pq, where p,q are different primes, or G is isomorphic to Q_8, or, for some prime p, G is isomorphic to one of the groups: N11p, C_p^3, elementary abelian group of order p^3.

بيژن طائري

محمدرضا ودادي

عليرضا عبداللهي , جواد باقريان , محمود بهبودي