A note on the size of the nilpotent residual in finite groups. (Q1925810)

Let \(G\) be a finite non-Abelian group with trivial Frattini subgroup, \(\text{In}(G)\) its group of inner automorphisms, \(G'\) its commutator subgroup and \(G^{\mathcal N}\) its nilpotent residual. \textit{Z. Halasi} and \textit{K. Podoski} [J. Algebra 319, No. 3, 893-896 (2008; Zbl 1177.20033)] and \textit{M. Herzog}, \textit{G. Kaplan} and \textit{A. Lev} [J. Algebra 320, No. 3, 980-986 (2008; Zbl 1157.20013)] have shown that \(\text{In}(G)\) of inner automorphisms of \(G\) is less than \(|\text{In}(G)|<|G'|^2\) while, under certain constraints on the order of \(G\), such as \(|G|\) being odd, \textit{S. Dolfi}, \textit{M. Herzog}, \textit{G. Kaplan} and \textit{A. Lev} [J. Group Theory 10, No. 3, 299-305 (2007; Zbl 1124.20013)] established the inequality \(|\text{In}(G)|<|G^{\mathcal N}|^2\). The main result of the present paper (Theorem 0.4) is: \(|\text{In}(G)|<|G'|\cdot|G^{\mathcal N}|\).