A note on automorphisms of finite \(p\)-groups. (English) Zbl 1354.20011
One of the longstanding conjectures related to automorphisms of finite \(p\)-groups – and now known to be false thanks to the work of J. González-Sánchez and A. Jaikin-Zapirain [Forum Math. Sigma 3, Article ID e7, 11 p. (2015; Zbl 1319.20019)] – was that \(|G|\) divides \(|\mathrm{Aut}(G)|\) whenever \(G\) is a finite noncyclic \(p\)-group of order at least \(p^3\).
Still, this leaves room for determining sufficient conditions on \(G\) such that \(|G|\) divides \(|\mathrm{Aut}(G)|\) and this is what the authors are concerned with in this paper. They prove that if \(G\) is a finite noncyclic \(p\)-group of order at least \(p^3\) and such that either a) \(G\) has an abelian maximal subgroup or b) \(Z(G)\) is elementary abelian and \(C_G(Z(\Phi(G)))\neq \Phi(G)\), then indeed \(|G|\) divides \(|\mathrm{Aut}(G)|\).
Still, this leaves room for determining sufficient conditions on \(G\) such that \(|G|\) divides \(|\mathrm{Aut}(G)|\) and this is what the authors are concerned with in this paper. They prove that if \(G\) is a finite noncyclic \(p\)-group of order at least \(p^3\) and such that either a) \(G\) has an abelian maximal subgroup or b) \(Z(G)\) is elementary abelian and \(C_G(Z(\Phi(G)))\neq \Phi(G)\), then indeed \(|G|\) divides \(|\mathrm{Aut}(G)|\).
Reviewer: Marian Deaconescu (Safat)
MSC:
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D45 | Automorphisms of abstract finite groups |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
