## Integral group ring of the Mathieu simple group $$M_{23}$$.(English)Zbl 1148.16027

In this paper the well-known Zassenhaus conjecture for the integral group ring of a finite group is studied, namely, that a torsion normalized unit is conjugate to some group element in the rational group algebra. This problem is still open, and by W. Kimmerle in 2005, the conjecture was proposed, that the prime graphs of the group and the group of normalized units coincide.
As a continuation of the authors’ papers [V. Bovdi, A. Konovalov, Lond. Math. Soc. Lect. Note Ser. 339, 237-245 (2007; Zbl 1120.16025) and V. Bovdi, A. B. Konovalov, S. Siciliano, Rend. Circ. Mat. Palermo (2) 56, No. 1, 125-136 (2007; Zbl 1125.16020)], in the present paper for the integral group ring of the Mathieu simple group $$M_{23}$$ the Kimmerle conjecture is verified. The method is due to Luther and Passi, and in addition to the verification of the conjecture, important information has been found concerning possible torsion units.

### 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

### Citations:

Zbl 1120.16025; Zbl 1125.16020

LAGUNA; GAP
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### References:

 [1] Artamonov V. A., Algebra Topology Geometry 27 pp 3– (1989) [2] Berman S. D., Ukrain. Mat. Ž. 7 pp 253– (1955) [3] Bleher F. M., LMS J. Comput. Math. 3 pp 274– (2000) · Zbl 0960.20004 [4] Bovdi V., J. Group Therapy 11 (1) pp 63– (2008) [5] Bovdi V., Groups St. Andrews 2005 1 pp 237– (2007) [6] Bovdi V., Publ. Math. Debrecen 65 (3) pp 291– (2004) [7] Bovdi V., LAGUNA – Lie AlGebras and UNits of group Algebras, Version 3.3.3. (2006) [8] Bovdi V., Rend. Circ. Mat. Palermo 56 (1) pp 125– (2007) · Zbl 1125.16020 [9] Cohn J. A., Canad. J. Math. 17 pp 583– (1965) · Zbl 0132.27404 [10] GAP – Groups, Algorithms, and Programming, Version 4.4.9. (2006) [11] Gorenstein D., The Classification of Finite Simple Groups 1 (1983) · Zbl 0609.20006 [12] Hertweck M., Proc. Edinb. Math. Soc. pp 1– (2005) [13] Hertweck M., Algebra Colloq. 13 (2) pp 329– (2006) · Zbl 1097.16009 [14] Höfert C., Groups, Rings and Group Rings 248 pp 243– (2006) · Zbl 1094.20027 [15] Kimmerle W., Groups, Rings and Algebras 420 pp 215– (2005) [16] Luthar I. S., Proc. Indian Acad. Sci. Math. Sci. 99 (1) pp 1– (1989) · Zbl 0678.16008 [17] Luthar I. S., Comm. Algebra 19 (8) pp 2353– (1991) · Zbl 0729.16021 [18] Marciniak Z., J. Number Theory 25 (3) pp 340– (1987) · Zbl 0611.16007 [19] Zassenhaus H., Studies in Mathematics pp 119– (1974)
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