Extended cut groups

TL;DR

Extended cut groups ($\rho(G)\le 1$) generalize cut groups and illuminate the center of the unit group $\mathcal{Z}(\mathrm{U}(\mathbb{Z}G))$ via $n_{\mathbb{R}}-n_{\mathbb{Q}}$. The authors develop conjugacy- and Wedderburn-based criteria for ecut groups, classify abelian and nilpotent ecut groups, and provide a complete classification for split metacyclic ecut groups using strong Shoda pairs. They give explicit results for dihedral and generalized quaternion ecut groups, present rank formulas for strongly monomial groups, and analyze monomial ecut groups with an emphasis on metacyclic families and a notable order-$1000$ example. Finally, the paper investigates the solvable ecut landscape, establishing prime-spectrum restrictions and illustrating the scope and limitations of the ecut property across solvable groups. Overall, the work deepens the structural understanding of central units in integral group rings and offers concrete classifications and computational tools for a broad spectrum of group classes.

Abstract

A finite group G is said to be a cut group if all central units in the integral group ring ZG are trivial. In this article, we extend the notion of cut groups, by introducing extended cut groups. We study the properties of extended cut groups analogous to those known for cut groups and also characterise some substantial classes of groups having the property of being extended cut. A complete classification of extended cut split metacyclic groups has been presented.

Theorems & Definitions (27)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 3.1: DMS05, Theorem 1 and Lemma 2, see also MP18, Theorem 5
• Theorem 3.2
• Corollary 3.3
• Theorem 3.4
• Theorem 3.5