The centers of discrete groups as stabilizers of Dark Matter

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.