On the number of subgroups of given structure in a finite \(p\)-group
As the title shows, the main interest of the author is to establish Anzahlsätze for finite \(p\)-groups. The main theorem of this paper will not be stated here as it needs too many preparatory definitions. Instead, to give a flavour of the results, I state only the last two ones. Let \(G\) be a finite \(p\)-group, let \(d(G)\) be the minimal number of generators of \(G\) and for \(1\leq i\leq d(G)\) let \(M_ i:= \{H< G\mid \Phi(G)\leq H\), \(| G:H|= p^ i\}\) and \(M_ i^ 0:=\{H\in M_ i\mid H\) is of maximal class\}. Proposition 11. If \(G\) is a 2-group and if \(G\) is not of maximal class, then for every \(i\) such that \(1\leq i\leq d(G)\) one has \(| M^ 0_ i |\equiv 0\pmod 4\). As a corollary, the following result of the author [Sib. Mat. Zh. 9, 1284-1306 (1968; Zbl 0315.20015)] is obtained: If a 2-group is not of maximal class, then the number of its subgroups of maximal class and of order at least 8 is a multiple of 4. Based on these facts, the author puts forward the following conjecture: If \(G\) is a group of order \(p^ m\), if \(1\leq i\leq d(G)\) and if \(i+ 1+p\leq m\), then \(| M^ 0_ i |\equiv 0\pmod {p^ 2}\).
scientific article; zbMATH DE number 687051
scientific article