Finite groups with its power automorphism groups having small indices

In this paper, a finite group G with |Aut(G): P(G)| = p or pq is determined, where P(G) is the power automorphism group of G, and p, q are distinct primes. Especially, we prove that a finite group G satisfies |Aut(G): P(G)| = pq if and only if Aut(G)/P(G) }~ S ₃. Also, some other classes of finite groups are investigated and classified, which are necessary for the proof of our main results. Partially supported by National Natural Science Foundation of China (Grant No. 10771132), SGRC (Grant No. GZ310), Shanghai Leading Academic Discipline Project (Grant No. J50101) and Science and Technology Foundation of Shanxi Province for Colleges (Grant No. 20081022)