Noninner automorphisms of order p for finite p-groups of restricted coclass

A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper, we give a lower bound for the coclass of finite nonabelian p-groups G having no noninner automorphism of order p leaving the Frattini subgoup Φ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (G)$$\end{document} elementwise fixed. As a consequence, the verification of the conjecture is reduced to the case of finite nonabelian p-groups G in which the coclass of G is greater than the minimum number of generators of G.