SU(3), U(1), O(3), etc what are they?

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I've heard them used before...some sort of symmetry, but I never figured out what it really is. Can someone please explain the concepts to me? thx
 

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HallsofIvy
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SU(3) is the "special unitary group of order 3": 3 by 3 complex matrices with determinant 1

O(3) is the "orthogonal group of order 3": 3 by 3 real matrices with determinant 1 or -1

U(1) is a bit puzzling! I would interpret U(n) as the "unitary group of order 3": n by n complex matrices with determinant 1 or -1. But "1 by 1 matrices" are just numbers. I guess U(1) would be the multiplicative group with members 1, -1, i, and -i!
 
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You will here them used to describe different types of standard model features.

U(1) corresponds to photon
SU(2) to W's and Z's
SU(3) to the gluons.
 

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