Let
$K$ be a field and
$G$ a finite group. Let
$G$ act on the rational
function field
$K\left(x\left(g\right):g\in G\right)$ by
$K$automorphisms
defined by
$g\cdot x\left(h\right)=x\left(gh\right)$
for any
$g$,
$h\in G$. Denote by
$K\left(G\right)$ the fixed field
$K{\left(x\left(g\right):g\in G\right)}^{G}$. Noether’s problem
then asks whether
$K\left(G\right)$
is rational (i.e., purely transcendental) over
$K$. The first main result of this
article is that
$K\left(G\right)$ is rational
over
$K$ for a certain class of
$p$groups having an abelian
subgroup of index
$p$. The
second main result is that
$K\left(G\right)$
is rational over
$K$ for
any group of order
${p}^{5}$
or
${p}^{6}$
(where
$p$
is an odd prime) having an abelian normal subgroup such that
its quotient group is cyclic. (In both theorems we assume that if
$charK\ne p$ then
$K$ contains a
primitive
${p}^{e}$th root
of unity, where
${p}^{e}$ is
the exponent of $G$.)
