Rationality problems for relation modules of dihedral groups

TL;DR

This paper investigates the module structure and rationality problem of relation modules associated with dihedral groups, providing insights into their algebraic properties and potential applications in group theory.

Contribution

It offers a detailed analysis of the module structure of R^{ab} for dihedral groups and addresses the rationality problem, which was previously not well-understood.

Findings

Characterization of the module structure of R^{ab}

Results on the rationality problem for dihedral groups

New techniques for analyzing relation modules

Abstract

Let D_n be the dihedral group of order 2n where n \ge 2, 1 \to R \to F \to D_n \to 1 be a free presentation of D_n. R^{ab}:=R/[R,R] becomes a \bm{Z}[D_n]-lattice. We will study the module structure and the rationality problem of R^{ab}.