I'm reading a book right now that goes into the adjoint representation of SU(N) to show that SU(N) has subgroups [tex]SU(N_1)[/tex], [tex]SU(N_2)[/tex], [tex]SU(N_3)[/tex], ... U(1), where [tex]N_1+N_2+N_3+...=N[/tex]. For example, SU(5) has subgroups SU(3), SU(2), and U(1).

My question is that since SU(N-1) is already subgroup, doesn't [tex]N_1=N-1[/tex], forcing [tex]N_2=1[/tex], and the rest of the N's zero?

Also, for U(1) to be a subgroup, you have to find a linear combination of generators (call it the generator [tex]T_0[/tex]) such that the new structure constant has a zero whenever one of its indices is 0? If the structure constant changes, isn't this a new group, and not a subgroup?

And why are the subgroups of SU(5) combined into the direct product group: SU(3)xSU(2)xU(1)?