Shokuev, V. N. Foundations of an enumeration theory for finite nilpotent groups. (Russian) Zbl 0858.20016 Zap. Nauchn. Semin. POMI 211, 174-183 (1994). The goal of this work is to show that the Möbius inversion is a very efficient tool for the solution of main problems of the enumeration theory for finite nilpotent groups. The key to this approach is in the author’s note [Ring and modules. Limit theorems of probability theory, No. 2, Leningrad 92-97 (1988; Zbl 0701.00009)], where he has computed the Möbius function on any subgroup of a finite \(p\)-group. Our enumeration method is based on Gaussian coefficients expressing the number of subgroups of order (index) \(p^n\) in an elementary (Abelian) \(p\)-group of order \(p^m\). For a \(p\)-group \(G\) of order \(p^m\), formulas for the number of subgroups of order \(p^{m-n}\), for the number of \(n\)-sequences of generators, for the number of solutions of the equation \(x^{p^n}=1\) are presented. Cited in 1 Review MSC: 20D15 Finite nilpotent groups, \(p\)-groups 20F05 Generators, relations, and presentations of groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D30 Series and lattices of subgroups Keywords:finite nilpotent groups; Möbius function; finite \(p\)-groups; number of subgroups; generators Citations:Zbl 0701.00009 PDFBibTeX XMLCite \textit{V. N. Shokuev}, Zap. Nauchn. Semin. POMI 211, 174--183 (1994; Zbl 0858.20016) Full Text: EuDML