On the subgroup lattice index problem in finite groups. (Q2017157)

In the theory of subgroup lattices of groups an important role is often played by the problem of determining the index of a subgroup. Following the approach introduced by \textit{M. De Falco, C. Musella, R. Schmidt} and the reviewer [Proc. Am. Math. Soc. 133, No. 4, 979-985 (2005; Zbl 1079.20042)], in this relevant paper the author characterizes in a finite group the subgroups of order \(p\) which are not \(p\)-atoms (where \(p\) is a prime number). Recall here that a \(p\)-atom of a finite lattice \(\mathcal L\) is essentially an atom of \(\mathcal L\) satisfying certain lattice properties which guarantee that for any isomorphism \(\varphi\) from \(\mathcal L\) onto the subgroup lattice of a (finite) group \(G\), the image under \(\varphi\) of any \(p\)-atom of \(\mathcal L\) is a subgroup of order \(p\) of \(G\). As an application of his main results on \(p\)-atoms, the author obtains, among many other interesting consequences, a lattice characterization of finite \(p\)-soluble groups. Moreover, a description is given of the projective closure of the class of finite \(p\)-nilpotent groups.