  Studying the general structure of the noncommutative (NC) local groups, we
prove a no-go theorem for NC gauge theories. According to this theorem, the
closure condition of the gauge algebra implies that: 1) the local NC $u(n)$
{\it algebra} only admits the irreducible n by n matrix-representation. Hence
the gauge fields are in n by n matrix form, while the matter fields {\it can
only be} in fundamental, adjoint or singlet states; 2) for any gauge group
consisting of several simple-group factors, the matter fields can transform
nontrivially under {\it at most two} NC group factors. In other words, the
matter fields cannot carry more than two NC gauge group charges. This no-go
theorem imposes strong restrictions on the NC version of the Standard Model and
in resolving the standing problem of charge quantization in noncommutative QED.
