The Cyclic Symmetries in the Representations of Unitary Discrete Subgroups

Dark matter may be stable because of a conserved $Z_N$ (cyclic) symmetry. Usually $N$ is assumed to be $2$, but it may also be larger than $2$. This $Z_N$ is usually assumed to be in a direct product with some other symmetry group. The full symmetry group of the theory is then $G = Z_N \times G'$. We suggest another possibility. Many discrete subgroups of $U(D)$, for any $D \ge 2$, have a non-trivial center $Z_N$, even if they are not the direct product of that $Z_N$ with some other group. When that happens, the irreducible representations (`irreps') of the group may either represent all the elements of that $Z_N$ by the unit matrix, or else they may represent that $Z_N$ faithfully. If ordinary matter is placed in a representation where $Z_N$ is represented by $1$, and dark matter is placed in irreps that represent $Z_N$ faithfully, then dark matter is stabilized by that $Z_N$. We have scanned all the discrete groups in the SmallGroups library with order $ \le 2000$ that are not the direct product of a cyclic group with some other group. We have determined their centers and whether they are subgroups of one or more groups $SU(D)$ or $U(D)$. We have found that very many groups, especially subgroups of $U(D)$ but not of $SU(D)$, have non-trivial centers $Z_N$, mostly with $N$ of the form $2^p \times 3^q$ but also with other values of $N$.