How do I "calculate" groups?

There are many weird symbols in physics like SO(8), SO(10), SU(5),
SU(5)*U(1), SU(4)*SU(2)*SU(2), SU(3)*SU(3)*SU(3) and so on. What do they
really mean? How do I "calculate" with them? I know that they are some kind
of groups but I do not have further details. Are there any good online
textbooks that teaches those SO(10) and so on -stuffs? I already know some
university algebra and physics, but I do not know those things. Please,
help.

--
Henri Heinonen - filosofian ylioppilas fysiikasta


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