On noninner automorphisms of finite p-groups that fix the Frattini subgroup elementwise

Let G be a finite p-group and let L*(G) = {a is an element of Z(Phi(G)) | a(2)p is an element of Z(G)}. In this paper we show that if L*(G) lies in the second center Z(2)(G) of G, then G admits a noninner automorphism of order p, when p is an odd prime, and order 2 or 4, when p = 2. Moreover, the automorphism can be chosen so that it induces the identity on the Frattini subgroup Phi(G). When p > 2, this reduces the verification of the well-known conjecture that states every finite nonabelian p-group G admits a noninner automorphism of order p to the case in which Z(2)*(G) (<)(not equal) L*(G), and C-G(Z(2)*(G)) = Phi(G), where Z(2)*(G) = {a is an element of Z(2)(G) | a is an element of Z(G)}. In addition, it follows that if G is a finite nonabelian p-group, p >= 2, such that Z(Phi(G)) is a cohomologically trivial G/Phi(G)-module, then G satisfies the above mentioned condition, and as a consequence we show that the order of G is at least p(8).