On the order of the automorphism group of a finite group

Let Gbea finite group, A (G) its group of automorphisms, and p a prime.Several authors have given upper bounds on the order o(A(G)) of A(G) in terms of o(G).Birkhoff and Hall [l, p. 499] suggested the problem of determining a lower bound for o((A(G)) in terms of o(G).For Abelian groups, this had already been done by Hilton [3].He proved that if pn\o(G), then pn~1(p-l)\o(A(G)) for Abelian groups G.The only result obtained in the general case is due to Herstein and Adney [2].They proved that if p2\o(G), then