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

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 ${\mathbb{Z}}_p[\mathrm{I}\mathrm{m}(\chi )][G]$-module come from ideal class groups, Artin $L$-functions and Iwasawa theory.