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Finite Class 2 Nilpotent and Heisenberg Groups

Dávid R. Szabó

TL;DR

The work provides a structural description of finite nilpotent groups of class at most $2$ via subdirect/central product decompositions and shows every such group embeds into a non-degenerate Heisenberg group determined by a suitable bilinear map. It develops alternating $\\ ext{Z}$-modules from group commutators, establishes a Darboux-type basis, and constructs polarised Heisenberg groups, enabling canonical embeddings. The results yield explicit decompositions and embeddings, with extended polarisation enabling cyclic-centre embeddings and a general method to realize all class-$2$ nilpotent groups as subgroups of Heisenberg-type groups. The topological motivation is to bound nilpotently Jordan properties for birational automorphism groups and homeomorphism groups, highlighting the practical impact of these algebraic structures on geometric and dynamical contexts.

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.

Finite Class 2 Nilpotent and Heisenberg Groups

TL;DR

The work provides a structural description of finite nilpotent groups of class at most via subdirect/central product decompositions and shows every such group embeds into a non-degenerate Heisenberg group determined by a suitable bilinear map. It develops alternating -modules from group commutators, establishes a Darboux-type basis, and constructs polarised Heisenberg groups, enabling canonical embeddings. The results yield explicit decompositions and embeddings, with extended polarisation enabling cyclic-centre embeddings and a general method to realize all class- nilpotent groups as subgroups of Heisenberg-type groups. The topological motivation is to bound nilpotently Jordan properties for birational automorphism groups and homeomorphism groups, highlighting the practical impact of these algebraic structures on geometric and dynamical contexts.

Abstract

We present a structural description of finite nilpotent groups of class at most using a specified number of subdirect and central products of -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.
Paper Structure (11 sections, 23 theorems, 18 equations)

This paper contains 11 sections, 23 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Notation
  4. Subdirect product decomposition
  5. Alternating modules
  6. Darboux generators and central product decomposition
  7. Complex structure
  8. Heisenberg groups
  9. Embedding into Heisenberg groups
  10. Extended polarisation and the induced embedding
  11. Improved embedding and the proof of \ref{['thm:mainHeisenberg']}

Key Result

Theorem 1.1

Every special $p$-group is a subdirect product of groups of the form: the central product of an extra-special $p$-group and an abelian group. Every extra-special $p$-group of order $p^{2n+1}$ is the central product of $n$ extra-special $p$-subgroups of order $p^3$. For every prime $p$, there are exa

Theorems & Definitions (69)

  • Theorem 1.1: Suzuki2
  • Theorem A
  • Theorem B
  • Corollary 1.2
  • Definition 1.3: guld2020finite2nilpotent
  • Theorem 1.4: guld2020finite2nilpotent based on prokhorov2016jordan
  • Theorem 1.5: ProkhorovShramov2014 or guld2019finiteDnilpotent for details
  • Theorem 1.6: nilpotentJordanHomeo
  • Theorem 1.7: CsikosMundetPyberSzabo
  • Definition 2.1
  • ...and 59 more