Finite p-groups with cyclic center have non-inner automorphisms of order p

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, under some conditions on an odd order finite p-group G with cyclic center, we prove that G exhibits a non-inner automorphism of order p. As a consequence, under certain conditions on a finite p-group G \((p>2),\) the conjecture is proved for all nilpotency classes except class 2 and maximal class. Moreover, we also settle the conjecture for some non-abelian finite 3-groups of coclass 3,  which is a pending case of the main result of Ruscitti et al. (Monatsh. Math. 183(4):679–697, 2016).