  Let $k$ be a field, $G$ be a finite group, $k(x(g):g\in G)$ be the rational
function field with the variables $x(g)$ where $g\in G$. The group $G$ acts on
$k(x(g):g\in G)$ by $k$-automorphisms where $h\cdot x(g)=x(hg)$ for all $h,g\in
G$. Let $k(G)$ be the fixed field defined by $k(G):=k(x(g):g\in G)^G=\{f\in
k(x(g):g\in G): h\cdot f=f$ for all $h\in G\}$. Noether's problem asks whether
the fixed field $k(G)$ is rational (= purely transcendental) over $k$. Let $m$
and $n$ be positive integers and assume that there is an integer $t$ such that
$t\in (\bm{Z}/m\bm{Z})^\times$ is of order $n$. Define a group
$G_{m,n}:=\langle\sigma,\tau:\sigma^m=\tau^n=1,\tau^{-1}\sigma\tau=\sigma^t\rangle$
$\simeq C_m \rtimes C_n$. We will find a sufficient condition to guarantee that
$k(G)$ is rational over $k$. As a result, it is shown that, for any positive
integer $n$, the set $S:=\{p: p$ is a prime number such that $\bm{C}(G_{p,n})$
is rational over $\bm{C} \}$ is of positive Dirichlet density; in particular,
$S$ is an infinite set.
