General linear groups as automorphism groups. (Q2911134)

If \(G\) is the direct product of \(d\) copies of the cyclic group \(C_p\) of prime order \(p\), then \(\Aut(G)\) is isomorphic to the general linear group \(\mathrm{GL}(d,p)\).NEWLINENEWLINE The authors address the interesting converse problem: is it true that if \(G\) is a finite \(p\)-group and if \(\Aut(G)\) is isomorphic to \(\mathrm{GL}(d,p)\), then \(G\) is an elementary Abelian \(p\)-group of rank \(d\)?NEWLINENEWLINE They give an affirmative answer in their main Theorem 2.5, and then, they even go further, to study finite \(p\)-groups \(G\) with a minimal number \(d\) of generators, and with \(|\Aut(G)|=|\mathrm{GL}(d,p)|\). As expected, this weaker condition gives weaker results.