On the conjecture of non-inner automorphisms of finite p-groups

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 if G is an odd order finite non-abelian monolithic p-group such that every maximal subgroup of G is non-abelian and [Z(M), g] ≤ Z(G) for every maximal subgroup M of G and g ∈ G M. Then G has a non-inner automorphism of order p leaving the Frattini subgroup Φ(G) elementwise fixed.