a characterization of a5 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, 597 (2022) 1-23.] showed that if g is non-solvable and o(g) = o(a5), then g ∼= a5. in this note, we prove that the equality o(g) = o(a5) does not hold for any finite solvable group g. consequently, up to isomorphism, a5 is determined by its average order.