  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 $K$. Let $p$ be any prime and let $G$ be a $p$-group of
exponent $p^e$. Assume also that {\rm (i)} char $K = p>0$, or {\rm (ii)} char
$K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. In this paper we
prove that if $G$ is any $p$-group of nilpotency class 2, which has the ABC
(Abelian-By-Cyclic) property, then $K(G)$ is rational over $K$. We also prove
the rationality of $K(G)$ over $K$ for two 3-generator $p$-groups $G$ of
arbitrary nilpotency class.
