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

Yutaka Konomi

TL;DR

This work analyzes the χ-parts of Z_p[Im(χ)][G]-modules arising from ideal class groups and Artin L-functions for the simple groups of orders 168, 504, and 360. It leverages Brauer induction to express χ as a sum of inductions from suitable subgroups and proves that e_χ^G M ≅ N^{(deg χ)} for a submodule N of M, valid for all χ except finitely many primes, with a precise formula for |N| when M is finite. The authors provide explicit subgroup data and constructive proofs for χ of various degrees across the three groups, complemented by extensive computer-assisted linear-algebra computations to realize the necessary elements in group rings and to verify surjectivity and kernel properties. A companion set of Magma scripts documents the computational backbone, enabling explicit Brauer inductions and module decompositions, and linking these χ-parts to class-group sizes and L-values via $L(0,χ,K/F_+)$ in appropriate CM-extensions. The results advance a concrete, computable framework for understanding Iwasawa-theoretic invariants in non-abelian simple-group contexts and illustrate how representation-theoretic decompositions translate into arithmetic information in Artin–L-function settings.

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 $χ$-part of a $\mathbb{Z}_p[\mathrm{Im}(χ)][G]$-module come from ideal class groups, Artin $L$-functions and Iwasawa theory.

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

TL;DR

This work analyzes the χ-parts of Z_p[Im(χ)][G]-modules arising from ideal class groups and Artin L-functions for the simple groups of orders 168, 504, and 360. It leverages Brauer induction to express χ as a sum of inductions from suitable subgroups and proves that e_χ^G M ≅ N^{(deg χ)} for a submodule N of M, valid for all χ except finitely many primes, with a precise formula for |N| when M is finite. The authors provide explicit subgroup data and constructive proofs for χ of various degrees across the three groups, complemented by extensive computer-assisted linear-algebra computations to realize the necessary elements in group rings and to verify surjectivity and kernel properties. A companion set of Magma scripts documents the computational backbone, enabling explicit Brauer inductions and module decompositions, and linking these χ-parts to class-group sizes and L-values via in appropriate CM-extensions. The results advance a concrete, computable framework for understanding Iwasawa-theoretic invariants in non-abelian simple-group contexts and illustrate how representation-theoretic decompositions translate into arithmetic information in Artin–L-function settings.

Abstract

Let be a prime and the ring of -adic integers. Let denote the simple group of order 168, 504 or 360. In this paper, we study the structure of the -part of a -module come from ideal class groups, Artin -functions and Iwasawa theory.

Paper Structure

This paper contains 27 sections, 16 theorems, 63 equations, 3 tables.

Table of Contents

  1. Introduction and the main results
  2. For the simple group of order 168
  3. The case of the characters of degree 3
  4. The case of the characters of degree 6
  5. The case of the characters of degree 7
  6. The case of the characters of degree 8
  7. For the simple group of order 504
  8. The case of the characters of degree 7
  9. The case of the characters of degree 8
  10. The case of the characters of degree 9
  11. For the simple group of order 360
  12. The case of the character of degree 5
  13. The case of the character of degree 9
  14. The case of the character of degree 10
  15. The case of the character of degree 8
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $G$ denote the simple group of order $168$, $504$ or $360$. Assume that $p$ does not divide $\# G$. If $7$ divides $\#G$, set $k=3\cdot 7$; otherwise, set $k=2^{2}\cdot 3\cdot 5$. Let $M$ be a $\mathbb{Z}_p[\zeta_k][G]$-module. For all $\chi \in \hbox{\rm Irr}(G)$, if $\#G\neq 360$ or $\deg(\chi such that the following two conditions hold except for a decidable finite number of primes: Here,

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Remark
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 9 more