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Foundations of an enumeration theory for finite nilpotent groups. (Russian) Zbl 0858.20016

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.

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

Citations:

Zbl 0701.00009
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