Enumerating finite class-2-nilpotent groups on 2 generators

TL;DR

This paper derives an explicit formula for the Dirichlet generating function counting finite class-2-nilpotent groups generated by at most two elements, providing a comprehensive enumeration method for these groups.

Contribution

It introduces a new explicit formula for the Dirichlet generating function that enumerates finite class-2-nilpotent groups generated by up to two elements.

Findings

Derived an explicit formula for the Dirichlet generating function

Provided a method to count finite class-2-nilpotent groups

Enhanced understanding of group enumeration in algebra

Abstract

We compute the numbers g(n,2,2) of nilpotent groups of order n, of class at most 2 generated by at most 2 generators, by giving an explicit formula for the Dirichlet generating function \sum_{n=1}^\infty g(n,2,2)n^{-s}.