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 $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ ($n\ge 3$) is the only simple Lie group with possible triangular anomaly. For this case, the anomaly coefficient has been explicitly computed in terms of the $n-1\mathrm{}$ parameters specifying irreducible representations of the group $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$. Various anomaly-free groups have been discussed, and it is argued that the best candidates for anomaly-free simple gauge groups are ${\mathrm{E}}_6$, $\mathrm{SO}\mathrm{}(4n+2)\mathrm{}$ ($n\ge 2$), and the vectorlike $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ ($n\ge 3$) theories.