On the conjecture of non-inner automorphisms of finite p-groups with a non-trivial abelian direct factor

Let p be a prime number. A longstanding conjecture asserts that every finite non-abelian p-group has a non-inner automorphism of order p. In this paper, we prove that the conjecture is true when a finite non-abelian p-group G has a non-trivial abelian direct factor. Moreover, we prove that the non-inner automorphism is central and fixes (G) elementwise. As a consequence, we prove that every group which is not purely non-abelian has a non-inner central automorphism of order p which fixes (G) elementwise.