A characterization of A_۵ by its average order

Let o(G) be the average order of a finite group G. M. Herzog, P. Longobardi and M. Maj [M. Herzog, P. Longobardi and M. Maj, Another criterion for solvability of finite groups, J. Algebra, ۵۹۷ (۲۰۲۲) ۱-۲۳.] showed that if G is non-solvable and o(G)=o(A_۵), then G\cong A_۵. In this note, we prove that the equality o(G)=o(A_۵) does not hold for any finite solvable group G. Consequently, up to isomorphism,A_۵ is determined by its average order.

کلیدواژه ها:

average order ، sum of element orders ، solvable group