[Paper Review] Integral group ring of the Mathieu simple group M23

Victor Bovdi, Alexander Konovalov|ArXiv.org|Dec 21, 2006

Finite Group Theory Research10 references4 citations

TL;DR

This paper confirms the Zassenhaus conjecture and Kimmerle's conjecture on prime graphs for the Mathieu simple group $M_{23}$ using the Luthar-Passi method. By analyzing partial augmentations of torsion units in the integral group ring $\mathbb{Z}M_{23}$, it proves that all torsion units of order not in \{12,24,28,56\} have orders realizable within $M_{23}$, and no units of orders 10, 21, 22, 33, 35, 46, 55, 69, 77, 115, 161, or 253 exist, thereby verifying the prime graph conjecture for $M_{23}$.

ABSTRACT

We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M23 using the Luthar-Passi method. This work is a continuation of the research that we carried out for Mathieu groups M11 and M12. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.

Motivation & Objective

• Address the classical Zassenhaus conjecture for the integral group ring of the sporadic simple Mathieu group $M_{23}$, extending prior work on $M_{11}$ and $M_{12}$.
• Verify Kimmerle's conjecture on the prime graph of $M_{23}$, which asserts that the prime graph of $V(\mathbb{Z}G)$ matches that of $G$.
• Investigate the structure of torsion units in $V(\mathbb{Z}M_{23})$ by determining constraints on their partial augmentations.
• Use representation-theoretic techniques to rule out the existence of torsion units of specific orders not present in $M_{23}$.
• Provide a complete classification of possible partial augmentation tuples for torsion units of orders 4, 6, 7, 8, 12, 24, 28, and 56.

Proposed method

• Apply the Luthar-Passi method to analyze torsion units in $V(\mathbb{Z}M_{23})$, focusing on partial augmentations with respect to conjugacy classes.
• Use character theory and the decomposition of group characters into irreducible constituents to compute the values $\mu_i(u,\chi_j,*)$, which must be non-negative integers.
• Employ Berman's theorem to enforce $\nu_1 = 0$ and the sum of all partial augmentations equal to 1, ensuring the unit is normalized.
• Systematically analyze all possible cases for units of orders not dividing $|M_{23}|$ or not realizable in $M_{23}$, using constraints from character values and integrality conditions.
• Use computational techniques and symbolic algebra to verify that no integral solutions exist for the systems of inequalities derived from the character values.
• Cross-validate results across multiple irreducible characters and conjugacy class structures to ensure consistency and completeness.

Experimental results

Research questions

• RQ1 Does every torsion unit in $V(\mathbb{Z}M_{23})$ have an order that matches the order of some element in $M_{23}$?
• RQ2 Are there any torsion units of order 10, 21, 22, 33, 35, 46, 55, 69, 77, 115, 161, or 253 in $V(\mathbb{Z}M_{23})$?
• RQ3 Is Kimmerle's prime graph conjecture satisfied for $M_{23}$, i.e., does $\pi(V(\mathbb{Z}M_{23})) = \pi(M_{23})$?
• RQ4 Can the Luthar-Passi method fully classify the partial augmentation tuples for torsion units of orders 4, 6, 7, 8, 12, 24, 28, and 56?
• RQ5 Are all torsion units of order 2, 3, 5, or 23 in $V(\mathbb{Z}M_{23})$ rationally conjugate to elements of $M_{23}$?

Key findings

• Units of order 12, 24, 28, or 56 in $V(\mathbb{Z}M_{23})$ are not ruled out by the analysis, but all other torsion units must have orders matching elements in $M_{23}$.
• No torsion units of order 10, 21, 22, 33, 35, 46, 55, 69, 77, 115, 161, or 253 exist in $V(\mathbb{Z}M_{23})$, as all corresponding systems of inequalities have no integral solutions with non-negative $\mu_i(u,\chi_j,*)$.
• For units of order 4, the partial augmentation tuple must be supported only on conjugacy classes $2a$ and $4a$, with $\nu_{2a}, \nu_{4a} \in \{(0,1), (-2,3), (2,-1)\}$.
• For units of order 6, the partial augmentations are supported only on $2a$, $3a$, and $6a$, with $\nu_{2a}, \nu_{3a}, \nu_{6a}$ taking values in a specific set of 15 integer triples.
• For units of order 7, the partial augmentations are supported only on $7a$ and $7b$, with $\nu_{7a}, \nu_{7b} \in \{(0,1), (2,-1), (1,0), (-1,2)\}$.
• Kimmerle's prime graph conjecture holds for $M_{23}$, as the prime graph of $V(\mathbb{Z}M_{23})$ matches that of $M_{23}$, confirmed by the absence of units of exotic orders.