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 $\mathrm{SU}(n)$ ($n\ensuremath{\ge}3$) is the only simple Lie group with possible triangular anomaly. For this case, the anomaly coefficient has been explicitly computed in terms of the $n\ensuremath{-}1$ parameters specifying irreducible representations of the group $\mathrm{SU}(n)$. Various anomaly-free groups have been discussed, and it is argued that the best candidates for anomaly-free simple gauge groups are ${\mathrm{E}}_{6}$, $\mathrm{SO}(4n+2)$ ($n\ensuremath{\ge}2$), and the vectorlike $\mathrm{SU}(n)$ ($n\ensuremath{\ge}3$) theories.