In GUTs one starts with some larger group, like $SU(5)$, which is then broken into smaller groups, for example
$$SU(5) ~\longrightarrow~ SU(3) \times SU(2) \times U(1)$$
This can be seen, for example, by looking at the Dynkin diagram for $SU(5)$: Removing one node leaves us with the Dynkin diagrams for $SU(3)$ and $SU(2)$.
My problem is understanding where $U(1)$ comes from. I've read several statements about that, but couldn't fit the puzzle pieces together. Dynkin diagrams are one the level of Lie algebras. Removing one node means we remove one generator. For example, in this 2010 handout from the the course Symmetries in Physics by Michael Flohr:
By removing a node, the rank of the subalgebra is reduced by one, and the simple roots are a subset of the original simple roots. On the level of the groups, we thus find, $$G=G_1\times G_2 \times U(1),$$ where the additional $U(1)$ factor comes from the left out Cartan generator. For example, $SU(n+m)$ can be reduced in this way into $$SU(n)\times SU(m)\times U(1).$$ This is the classical ansatz for a GUT: $SU(5)$ gets broken into $$SU(3)\times SU(2)\times U(1).$$
- In this paper by John Baez it is claimed that $SU(3) \times SU(2) \times U(1)$ is no subgroup of $SU(5)$ and some homomorphism is used the justify why $U(1)$ appears.
In Lie Algebras In Particle Physics: from Isospin To Unified Theories Georgi writes:
"The Cartan generator that have been left out generate $U(1)$ factors"
and my problem is understanding why this is the case.
Any ideas to clarify this would be awesome!