Why do we want GUT groups to be simple (rather than, say, semisimple)? I've heard before that when looking for grand unified gauge groups, with a single gauge coupling, we want to use simple groups, in the group-theoretic sense of having no non-trivial normal subgroups. What is the reason for this? To me it doesn't seem as though there is a problem with having a GUT group (like $SU(3) \times SU(3)$ for instance) which is not simple, but just giving the group a single gauge coupling. Can anybody shed any light on this?
 A: If you don't impose some sort of simplicity restriction on the grand unified group, then the Standard Model gauge group $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ would already classify as a GUT group, making the search for a GUT pointless. There is no problem as such with non-simple gauge groups, but they just don't correspond to what we call "grand unified theories", and choosing "a single gauge coupling" isn't really what GUTs are about. One of the best motivations for GUTs is that we expect them to naturally solve the problem of the quantization of electric charge, in particular why the hadron/quark and the lepton balance each other so nicely. In order to solve this, you must forbid such things as the electromagnetic $\mathrm{U}(1)$ showing up as its own factor, so that its representation is induced by the stricter representation theory of the grand unified group, and not chosen as input as in the Standard Model.
This terminology is slightly at odds with saying that the electroweak $\mathrm{SU}(2)\times\mathrm{U}(1)$ "unifies" the weak $\mathrm{SU}(2)$ and the electromagnetic $\mathrm{U}(1)$, but there's not much we can do about that. Note, however, that at least the electromagnetic $\mathrm{U}(1)$ and the weak $\mathrm{SU}(2)$ are not the factors of the electroweak group - the electromagnetic $\mathrm{U}(1)$ embeds into the electroweak $\mathrm{SU}(2)\times\mathrm{U}(1)$ as a mixture of the isospin and the weak hypercharge.
