Finite <i>p</i> -groups satisfying a weak chain condition (Q6594893)

Let \(p\) be a prime number and let \(G\) be a finite group of order \(p^n\). Then there exists a unique \(0\leq t\leq n-2\) such that all subgroups of index \(p^t\) in \(G\) are abelian but the same does not hold for its subgroups of index \(p^{t-1}\). With respect to such \(t\), the group \(G\) is said to be an \textit{\(\mathcal{A}_t\)-group}. Being an \(\mathcal{A}_0\)-group is the same as being abelian, while being an \(\mathcal{A}_1\)-group is equivalent to being minimal non-abelian.\N\NLet now \(H\) be a subgroup of \(G\). Then \(H\) is an \(\mathcal{A}_u\)-group for some integer \(u\leq t\) and, if \(|G:H|\geq p^t\), then \(H\) is abelian. Let now \(1\leq i\leq t-1\) be an integer and assume that \(|G:H|=p^i\). Then \(H\) is a \textit{chain subgroup} of \(G\) if \(H\) is an \(\mathcal{A}_{t-i}\)-group; otherwise it is named a \textit{non-chain subgroup}. Though \(G\) has chain subgroups of any index, the presence of non-chain subgroups is not guaranteed.\N\NIn the Introduction, the following three problems are proposed:\N\begin{itemize}\N\item[1.] Classify the finite \(p\)-groups that do not contain non-chain subgroups.\N\item[2.] Classify the finite \(p\)-groups that contain non-chain subgroups of any index and any type (in the sense of Figure 1).\N\item[3.] Classify the finite \(p\)-groups all of whose non-chain subgroups are abelian.\N\end{itemize}\NProblem 1 being already solved (see [\textit{L. Zhang} and \textit{H. Qu}, J. Algebra Appl. 13, No. 4, Article ID 1350137, 5 p. (2014; Zbl 1325.20013); \textit{Q. Zhang}, Algebra Colloq. 26, No. 1, 1--8 (2019; Zbl 1490.20018)]), the article addresses Problems 2 and 3.\N\NProblem 2 is solved in Section 5 partitioning the finite \(p\)-groups into \(\mathcal{A}_t\)-groups, as \(t\) varies. In this case \(t\geq 2\) and Lemma 5.1 reduces Problem 2 to the consideration of non-chain subgroups of index \(p\). The cases \(t=2\) and \(t=3\) are handled in [\textit{Q. Zhang} et al., Algebra Colloq. 15, No. 1, 167--180 (2008; Zbl 1153.20018); Commun. Math. Stat. 3, No. 1, 69--162 (2015; Zbl 1322.20012)], while when \(t\geq 4\) Problem 2 is solved in Theorem 5.4.\N\NProblem 3 is solved in Section 3, with the additional assumption that the abelian non-chain subgroups of a fixed index are cyclic. In particular, Corollary 3.4 shows that this is necessary and sufficient for all non-chain subgroups to be cyclic, which is equivalent to the ambient group being a \(2\)-group of maximal class. Section 4 studies \(\mathcal{A}_t\)-groups from the point of view of the number of non-chain subgroups of a given index.