On one generalization of finite \(\mathfrak{U}\)-critical groups (Q2797019)

This paper considers only finite groups. A proper subgroup \(H\) of a group \(G\) is called \(\mathbb{P}\)-subnormal in \(G\) if there exists a chain of subgroups \(H=H_0 < H_1< \cdots <H_n=G\) such that \(|H_i : H_{i-1}|\) is a prime, for \(i=1, \ldots, n\). \(H\) is said to be \(\mathbb{P}\)-abnormal in \(G\) if \(|L : K|\) is not a prime, whenever \(K , L\) are subgroups of \(G\) such that \(H \leq K\leq L\).NEWLINENEWLINEIn the universe of all soluble groups, the concepts of \(\mathbb{P}\)-subnormality and \(\mathbb{P}\)-abnormality coincide with the more classical ones of \(\mathfrak{F}\)-subnormality and \(\mathfrak{F}\)-abnormality, respectively, for the class \(\mathfrak{F}=\mathfrak{U}\) of all supersoluble groups.NEWLINENEWLINEGiven a saturated formation \(\mathfrak{F}\), a subgroup \(H\) of a group \(G\) is said to be \(\mathfrak{F}\) -subnormal in \(G\) if a chain of subgroups \(H=H_0 < H_1< \cdots <H_n=G\) exists such that \(H_{i-1}\) is maximal in \(H_{i}\), and \(H_i/ (H_{i-1})_{H_{i}} \in \mathfrak{F}\), for \(i=1, \ldots, n\). \(H\) is called \(\mathfrak{F}\)-abnormal in \(G\) if \(L/K_L \not\in \mathfrak{F}\), whenever \(H \leq K\leq L \leq G\). (\(U_X\) denotes the core in \(X\) of \(U\).) A group \(G \not\in \mathfrak{F}\) is said to be an \(E_{\mathfrak{F}}\)-group if every non-trivial subgroup of \(G\) is either \(\mathfrak{F}\)-subnormal, or \(\mathfrak{F}\)-abnormal.NEWLINENEWLINEThe complete description of all \(E_{\mathfrak{F}}\)-groups was given by \textit{G. Ebert} and \textit{S. Bauman} [J. Algebra 36, 287--293 (1975; Zbl 0314.20019)] in the case when \(\mathfrak{F}=\mathfrak{N}\) is the class of all nilpotent groups, and when \(\mathfrak{F}\) is the class of all soluble \(p\)-nilpotent groups, for an odd prime \(p\). Other authors also considered this problem for classes \(\mathfrak{F}\) for which every \(\mathfrak{F}\)-critical group (a group not in \(\mathfrak{F}\) all whose proper subgroups are in \(\mathfrak{F}\)) is a Schmidt group, i.e., an \(\mathfrak{N}\)-critical group. The main theorem of the paper under review (Theorem A) describes all non-supersoluble groups which are \(E_{\mathfrak{U}}\)-groups, \(\mathfrak{U}\) the class of all supersoluble groups.