[Paper Review] Kimmerle conjecture for the Held and O'Nan sporadic simple groups

Victor Bovdi, A. Grishkov+1 more|ArXiv.org|Feb 13, 2009

Finite Group Theory Research20 references3 citations

TL;DR

This paper confirms the Kimmerle conjecture for the Held and O'Nan sporadic simple groups using the Luthar–Passi method, analyzing torsion units in their integral group rings. It proves the non-existence of normalized torsion units of orders 34, 35, 51, 85, and 119 in the Held group, and shows that all such units in both groups are rationally conjugate to elements of the group except possibly for orders 33 and 57.

ABSTRACT

Using the Luthar--Passi method, we investigate the Zassenhaus and Kimmerle conjectures for normalized unit groups of integral group rings of the Held and O'Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held simple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture.

Motivation & Objective

• To verify the Kimmerle conjecture for the Held and O’Nan sporadic simple groups by analyzing torsion units in their integral group rings.
• To investigate the structure of normalized unit groups of integral group rings for these groups using character-theoretic and computational techniques.
• To determine the existence or non-existence of torsion units of various orders, especially those relevant to the prime graph condition in the Kimmerle conjecture.
• To derive constraints on partial augmentations of torsion units, supporting further investigation of the Zassenhaus conjecture.
• To extend the applicability of the Luthar–Passi method to previously unverified sporadic simple groups, contributing to the broader classification of torsion units in integral group rings.

Proposed method

• Employing the Luthar–Passi method to analyze partial augmentations of torsion units in the normalized unit group $V(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$.
• Using ordinary and Brauer characters, particularly $p$-Brauer characters for $p=3,7$, to derive constraints on the partial augmentations of torsion units.
• Applying the Berman–Higman theorem to enforce $\nu_1 = 0$, ensuring the sum of partial augmentations over non-trivial conjugacy classes equals 1.
• Formulating systems of inequalities involving the $\mu_i(u,\chi,p)$ values derived from character values and partial augmentations to test integrality and non-negativity.
• Leveraging the GAP package LAGUNA for computational algebra to speed up character calculations and validate theoretical constraints.
• Using the prime graph $\pi(G)$ and the condition $\pi(G) = \pi(V(\mathbb{Z}G))$ as the central criterion for verifying the Kimmerle conjecture.

Experimental results

Research questions

• RQ1 Does the Kimmerle conjecture hold for the Held sporadic simple group?
• RQ2 Are there normalized torsion units of order 34, 35, 51, 85, or 119 in $V(\mathbb{Z}G)$ for the Held group?
• RQ3 What are the constraints on partial augmentations of torsion units of order 3, 5, 17, 22, 35, or 31 in $V(\mathbb{Z}G)$ for the Held and O’Nan groups?
• RQ4 Can the Luthar–Passi method be successfully applied to confirm the Kimmerle conjecture for previously unverified sporadic simple groups?
• RQ5 What is the status of the Zassenhaus conjecture for the O’Nan group, particularly regarding units of orders 33 and 57?

Key findings

• The Kimmerle conjecture is confirmed for the Held sporadic simple group, as $\pi(G) = \pi(V(\mathbb{Z}G))$ holds.
• There are no normalized torsion units of order 34, 35, 51, 85, or 119 in $V(\mathbb{Z}G)$ for the Held group.
• For units of order 5 in the Held group, the paper proves rational conjugacy to an element of $G$, confirming part of the Zassenhaus conjecture.
• For units of order 3 in the Held group, the partial augmentations $\nu_{3a}$ and $\nu_{3b}$ satisfy $-4 \leq \nu_{3a} \leq 5$ and $\nu_{3a} + \nu_{3b} = 1$, with all other $\nu_{kx} = 0$.
• For units of order 17 in the Held group, the partial augmentations are constrained to a specific set of integer solutions, though the full set is not listed in the abstract.
• For the O’Nan group, the paper shows that normalized torsion units of orders 33 and 57 are not ruled out, but all other orders relevant to the prime graph are excluded.