On finite \(p\)-groups with Abelian automorphism group. (Q2852573)

Let \(G\) be a finite group and let \(\Aut(G)\) denote its automorphism group. An old and intriguing problem is to say something about \(G\) knowing that \(\Aut(G)\) is Abelian. Of course, the problem is quickly reduced to that of \(G\) being a \(p\)-group and the authors offer a generous introduction to the related literature.NEWLINENEWLINE The authors show that when \(G\) is a \(p\)-group and \(\Aut(G)\) is elementary Abelian, then either \(\Phi(G)=G'\) is elementary Abelian, or \(\Phi(G)=Z(G)\) is elementary Abelian and then construct examples (for \(p\) an odd prime) to show that exactly one of the two equalities holds. Examples are also constructed (for odd primes \(p\)) of \(p\)-groups \(G\) with \(\Aut(G)\) Abelian and satisfying exactly one of the conditions \(G'=Z(G)<\Phi(G)\), \(G'<Z(G)<\Phi(G)\).NEWLINENEWLINE As the authors remark, their examples suggest that general results on the structure of a finite \(p\)-group \(G\) having Abelian automorphism group \(\Aut(G)\) will be hard to obtain.