On group rings of the simple group of order 168, 504 or 360 and their modules

TL;DR

This paper investigates the structure of modules over group rings of specific simple groups of orders 168, 360, and 504, focusing on their $ ext{chi}$-parts related to ideal class groups, Artin $L$-functions, and Iwasawa theory.

Contribution

It provides new insights into the module structure over group rings of these simple groups, connecting algebraic and number-theoretic aspects.

Findings

Characterizes the $ ext{chi}$-parts of modules over these group rings.

Links module structures to ideal class groups and Artin $L$-functions.

Advances understanding in Iwasawa theory contexts.

Abstract

Let $p$ be a prime and $\mathbb{Z}_p$ the ring of $p$-adic integers. Let $G$ denote the simple group of order 168, 504 or 360. In this paper, we study the structure of the $\chi$-part of a $\mathbb{Z}_p[\mathrm{Im}(\chi)][G]$-module come from ideal class groups, Artin $L$-functions and Iwasawa theory.