Finite \(p\)-groups all of whose subgroups of index \(p^3\) are Abelian. (Q2343045)

The finite nonabelian groups in which all proper subgroups are abelian were determined by \textit{G. A. Miller} and \textit{H. C. Moreno} [Trans. Am. Math. Soc. 4, 398-404 (1903; JFM 34.0173.01)]. (For the case of \(p\)-groups, see also the paper by \textit{L. Redei} [Comment. Math. Helv. 20, 225-264 (1947; Zbl 0035.01503)].) Later \textit{V. A. Sheriev} [in: Sem. algebraiceskie Sist. 2, 25-53 (1970; Zbl 0251.20014); ibid. 2, 54-76 (1970; Zbl 0251.20015)] was able to extend this classification to the finite \(p\)-groups in which all subgroups of index \(p^2\) are abelian. This classification has been revisited by several authors, in particular by \textit{Q. Zhang} et al. [Algebra Colloq. 15, No. 1, 167-180 (2008; Zbl 1153.20018)]. The main goal of the paper under review is to classify the finite \(p\)-groups in which all subgroups of index \(p^3\) are abelian. The classification is spread out over several theorems, which deal with a number of different cases; some of them require lengthy, involved calculations. The total number of isomorphism classes is \(222\).