Defining Group Multiplication in Particle Physics

[quantumfireball]

Everyone must be familiar with U(1),SU(2) and SU(3) Lie groups in particle physics .
But how does one define the multiplication of two groups of different dimensions
aka SU(2) X U(1) or SU(3) X SU(2) X U(1).

 

[matt grime]

You aren't multiplying them. That is the direct product of the groups. Give two groups the direct product GxH is the set of all pairs (g,h) g in G, h in H with the natural composition

(g,h)(g',h')=(gg',hh')

 

[quasar987]

It's also the direct sum. If ever you encounter the notation $$\bigoplus_{i=1}^n G_i$$, this is what it means. Take the cartesian product of the G_i and give them a group structure by defining the binary operation as in matt grime's post. (g1,...,gn)(g1',...,gn'1)=(g1g1',...,gngn')