Rationality problem of three-dimensional purely monomial group actions: the last case

A $k$-automorphism $\sigma$ of the rational function field $k(x_1,\dots ,x_n)$ is called purely monomial if $\sigma$ sends every variable $x_i$ to a monic Laurent monomial in the variables $x_1,\dots ,x_n$. Let $G$ be a finite subgroup of purely monomial $k$-automorphisms of $k(x_1,\dots ,x_n)$. The rationality problem of the $G$-action is the problem of whether the $G$-fixed field ${{k}\left({{x_1},\dots,{x_n}}\right)^{G}}$ is $k$-rational, i.e., purely transcendental over $k$, or not. In 1994, M. Hajja and M. Kang gave a positive answer for the rationality problem of the three-dimensional purely monomial group actions except one case. We show that the remaining case is also affirmative.