On the existence of a noninner automorphism of order \(p\) for \(p\)-groups. (Q2784888)

The authors study automorphism groups of finite \(p\)-groups in connection with an open problem of Ya.~Berkovich: prove that any finite non-Abelian \(p\)-group \(G\) possesses a noninner automorphism of order \(p\). Although the order of \(\text{Out}(G)\) is divisible by \(p\) (because of a known theorem of Gaschütz) this does not imply the existence of such an automorphism.NEWLINENEWLINENEWLINEThe main result of the paper is the following. Let \(G\) be a regular non-Abelian \(p\)-group and \(H\) a subgroup of index \(p\). If there exists an element \(x\in Z(H)\) of order \(p\) such that \(x\not=[a,\nu]\), where \(a\not\in H\), \(\nu\in Z(H)\), then \(G\) possesses a noninner automorphism of order \(p\). It is also shown that any group of order \(p^n\) (\(n<7\), \(p>3\) a prime) admits a noninner automorphism of order \(p\).