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

A k-automorphism σ of the rational function field k(x1, . . . , xn) is called purely monomial if σ sends every variable xi to a monic Laurent monomial in the variables x1, . . . , xn. Let G be a finite subgroup of purely monomial k-automorphisms of k(x1, . . . , xn). The rationality problem of the G-action is the problem of whether the G-fixed field k(x1, . . . , xn) 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.