Noether’s Problem for p-Groups with Abelian Normal Subgroups and Central p-Powers

This paper addresses Noether’s problem for p-groups G, having an abelian normal subgroup of index p, under the condition Gp={gp:g∈G}≤Z(G)—the center of G. We prove that the fixed field K(G)=K(x(g):g∈G)G is rational over K in such cases, focusing on both the classification and structural analysis of these groups. Our results extend existing work by removing restrictive assumptions and providing a refined understanding of p-groups and their representations.