The centers of discrete groups as stabilizers of Dark Matter

TL;DR

This paper explores how the centers of discrete groups, particularly those not decomposable into direct products, can serve as stabilizers for Dark Matter in theoretical models, expanding beyond the common $Z_2$ symmetry.

Contribution

It systematically analyzes discrete groups of order less than 2001 to identify non-decomposable groups with non-trivial centers suitable for Dark Matter stabilization.

Findings

Many small discrete groups have non-trivial centers.

Some groups cannot be expressed as a direct product, yet have stabilizing centers.

Potential new symmetries for Dark Matter stabilization identified.

Abstract

The most usual option to stabilize Dark Matter (DM) is a $Z_2$ symmetry. In general, though, DM may be stabilized by any $Z_N$ with $N \ge 2$. We consider the way $Z_N$ is a subgroup of the internal-symmetry group $G$ of a model; we entertain the possibility that $Z_N$ is the center of $G$, yet $G$ is not of the form $Z_N \times G^\prime$, where $G^\prime$ is a group smaller (i.e. of lower order) than $G$. We examine all the discrete groups of order smaller than 2001 and we find that many of them cannot be written as the direct product of a cyclic group and some other group, yet they have a non-trivial center that might be used in Model Building to stabilize DM.