Central automorphism groups fixing the center element-wise. (Q2920206)

Let \(G\) be a finite \(p\)-group, let \(Z\) denote its center and let \(e=p^n\) be the exponent of \(Z\). Finally, let \(G^e=\langle g^e\mid g\in G\rangle\).NEWLINENEWLINE The author addresses the following natural (and, quite important in several ``technical'' contexts) question: under what conditions on \(G\) is it true that \textit{every} automorphism of \(G\) which acts trivially on \(G/Z\) also acts trivially on \(Z\)? An immediate \textit{sufficient} condition is that \(Z\leqslant G'\).NEWLINENEWLINE An elegant answer is obtained by the author in the main theorem of this nice short note: a necessary and sufficient condition is that \(Z\leqslant G^eG'\).