# A structure theorem for product sets in extra special groups

## TL;DR

This paper provides a new, robust proof of a structure theorem for product sets in extra special groups, extending previous results from Heisenberg groups to a broader class of groups and quasigroups.

## Contribution

The authors introduce a new proof technique that generalizes the structure theorem for product sets from Heisenberg groups to all extra special groups and certain quasigroups.

## Key findings

- Product sets in large bricks of extra special groups contain many cosets of the center.
- The new proof extends the theorem's applicability beyond Heisenberg groups.
- The approach is robust and adaptable to a wider class of algebraic structures.

## Abstract

Hegyv\'ari and Hennecart showed that if $B$ is a sufficiently large brick of a Heisenberg group, then the product set $B\cdot B$ contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.