Gauge groups without triangular anomaly

Suppose that G is a simple gauge group governing a unified gauge theory. We shall then prove that the existence or absence of the triangular anomaly is equivalent to the same question for symmetrized third-order Casimir invariants of G. Consequently, we show that the group SU (n) (n3) is the only simple Lie group with possible triangular anomaly. For this case, the anomaly coefficient has been explicitly computed in terms of the n-1 parameters specifying irreducible representations of the group SU (n). Various anomaly-free groups have been discussed, and it is argued that the best candidates for anomaly-free simple gauge groups are E₆, SO (4n+2) (n2), and the vectorlike SU (n) (n3) theories.