On the order of the Sylow subgroups of the automorphism group of a finite group

Given any finite groupG, we wish to determine a relationship between the highest power of a primepdividing the order ofG, denoted by |G|p, and |A(G)|p, whereA(G)is the automorphism group ofG. It was shown by Herstein and Adney [8] that |A(G)|p≧ p whenever|G|p= ≧P2. Later Scott [16] showed thatA(G)p≧P2. For the special case whereGis abelian, Hilton [9] proved that Adney [1] showed that this result holds if a Sylow p-subgroup ofGis abelian, and gave an example where|G|p= p4and|A(G)|P=p2. We are able to show in Theorem 4.5 that, if|G|p= ≧ p5, then|A(G)| = ≧ p3.