Finite Class 2 Nilpotent and Heisenberg Groups

TL;DR

This paper provides a structural classification of finite class 2 nilpotent groups, showing they can be embedded into certain Heisenberg groups, with implications for topological automorphism groups.

Contribution

It offers a new structural description of finite class 2 nilpotent groups and demonstrates their isomorphism to subgroups of specific Heisenberg groups.

Findings

Finite class 2 nilpotent groups can be constructed from subdirect and central products.

All such groups are isomorphic to subgroups of certain Heisenberg groups.

Results have applications in bounding automorphism groups of varieties and manifolds.

Abstract

We present a structural description of finite nilpotent groups of class at most $2$ using a specified number of subdirect and central products of $2$-generated such groups. As a corollary, we show that all of these groups are isomorphic to a subgroup of a Heisenberg group satisfying certain properties. The motivation for these results is of topological nature as they can be used to give lower bounds to the nilpotently Jordan property of the birational automorphism group of varieties and the homeomorphism group of compact manifolds.