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