Color embeddings, charge assignments, and proton stability in unified gauge theories

Three problems in hypothetical unified theories of electromagnetic, weak, and strong interactions are discussed here. First, the problem of embedding color in any simple gauge group is solved, and a complete classification of theories where the fermion color is restricted to $1^c$, $3^c$, and ${\overline{3}}^c$ of $\mathrm{S}{\mathrm{U}}_3^c$ is given. Generalizations are also indicated. Second, an unbroken ${\mathrm{U}}_1$ generated by electric charge is embedded into each of the above theories and the charge assignments analyzed. Third, the general problem of stabilizing the proton by a suitable atomic mass number $A$ is studied for the same set of theories. It is always possible to define $A$ if the gauge group is not too small. In many of these theories there must be fermions with weird values of $A$: leptons with $A\ne 0$ and quarks with $\mathrm{A}\ne \frac{1}{3}$. Examples are discussed. Some future directions of research are indicated.