A direct proof of Müller’s result on automorphisms of finite p-groups

Müller proved that, if G is a finite p-group which is neither elementary abelian nor extra-special, then AutΦ(G)/Inn(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Aut}\,}}^{\Phi }(G) / {{\,\mathrm{Inn}\,}}(G)$$\end{document} is a non-trivial normal p-subgroup of the group of outer automorphisms of G. Here AutΦ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Aut}\,}}^{\Phi }(G)$$\end{document} denotes the group of all automorphisms of G that centralize the Frattini quotient G/Φ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G/\Phi (G)$$\end{document} of G. We give a new direct proof, which avoids, in particular, the use of cohomological considerations.