I always found it a curiousity that in the symmetry groups of the known fundamental forces we find the nice arithmetic progression $1,2,3$: first there is $\DeclareMathOperator{\U}{U}\DeclareMathOperator{\SU}{SU}\U(\color{red} 1)$, then $\SU(\color{red}2)$, and finally $\SU(\color{red}3)$.
Questions:
- Is this merely a coincidence?
- Were there reasons to believe, that after having found $\U(1)$ and $\SU(2)$ symmetries, that the next symmetry to look out for is $\SU(3)$?
- Have people considered $\SU(4)$ symmetries, or is there a reason it stops after $3$?
I have heard of $\SU(5)$ theories, but those seem motivated not by the arithmetic progression, but by the fact that $\SU(5)$ contains $\U(1)\times\SU(2)\times\SU(3)$.
