Computing the decomposition group of a zero-dimensional ideal by elimination method

In this note, we show that the decomposition group $Dec(I)$ of a zero-dimensional radical ideal $I$ in $\mathbf{K}[x_1,\dots ,x_n]$ can be represented as the direct sum of several symmetric groups of polynomials based upon using Gr"{o}bner bases. The new method makes a theoretical contribution to discuss the decomposition group of $I$ by using Computer Algebra without considering the complexity. As one application, we also present an approach to yield new triangular sets in computing triangular decomposition of polynomial sets $\mathbb{P}$ if $Dec(<\mathbb{P}>)$ is known.