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Integral group ring of the Mathieu simple group \(M_{12}\). (English) Zbl 1125.16020

The Zassenhaus conjecture states that normalized torsion units of the integral group ring of a finite group are conjugate in the rational group algebra to an element of the group. This conjecture has been considered by many researchers over decades and also has some variations. A recent one is due to W. Kimmerle: the prime graph of the group and the group of normalized units coincide (the prime graph of a group has the prime divisors of the orders of torsion elements as vertices, and has edges between two primes if there is an element of order the product of the primes in the group). This weaker conjecture was verified by W. Kimmerle for finite Frobenius and solvable groups.
In this paper, the authors establish Kimmerle’s conjecture for the Mathieu simple group \(M_{12}\). The tools of the proof are known restrictions on partial augmentations of torsion units in terms of characters, used for investigations concerning the Zassenhaus conjecture by, for instance, Z. Marciniak, J. Ritter, S. K. Sehgal, and A. Weiss, [J. Number Theory 25, 340-352 (1987; Zbl 0611.16007)] and I. S. Luthar and I. B. S. Passi, [Proc. Indian Acad. Sci., Math. Sci. 99, No. 1, 1-5 (1989; Zbl 0678.16008)].

MSC:

16U60 Units, groups of units (associative rings and algebras)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
20D08 Simple groups: sporadic groups

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