On Heisenberg groups

TL;DR

This paper demonstrates that under mild conditions, any class 2 nilpotent group can be represented as a Heisenberg group extension with a bimultiplicative 2-cocycle, unifying a broad class of such groups.

Contribution

It proves that all class 2 nilpotent groups are equivalent to a Heisenberg group extension with a bimultiplicative 2-cocycle under mild conditions.

Findings

Any class 2 nilpotent group can be expressed as a Heisenberg group extension.

The 2-cocycle in such extensions can be chosen to be bimultiplicative.

This unifies the structure of class 2 nilpotent groups under a common framework.

Abstract

It is known that an abelian group $A$ and a $2$-cocycle $c:A \times A \to C$ yield a group ${\mathscr{H}}(A,C,c)$ which we call a Heisenberg group. This group, a central extension of $A$, is the archetype of a class~$2$ nilpotent group. In this note, we prove that under mild conditions, any class~$2$ nilpotent group $G$ is equivalent as an extension of $G/[G,G]$ to a Heisenberg group ${\mathscr{H}}(G/[G,G], [G,G], c')$ whose $2$-cocycle $c'$ is bimultiplicative.