## Zassenhaus conjecture for $$A_6$$.(English)Zbl 1149.16027

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 few non-solvable groups are known to satisfy the conjecture. Recent papers, for instance by V. Bovdi, A. Konovalov, [Lond. Math. Soc. Lect. Note Ser. 339, 237-245 (2007; Zbl 1120.16025)], and V. A. Bovdi, A. B. Konovalov, S. Siciliano, [Rend. Circ. Mat. Palermo (2) 56, No. 1, 125-136 (2007; Zbl 1125.16020)], and the present paper use the method due to I. S. Luthar, I. B. S. Passi, [Proc. Indian Acad. Sci., Math. Sci. 99, No. 1, 1-5 (1989; Zbl 0678.16008)], for verification of the conjecture for particular simple groups. In this paper the author establishes the conjecture for the alternating group of degree 6.

### 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 20C30 Representations of finite symmetric groups

### Citations:

Zbl 1120.16025; Zbl 1125.16020; Zbl 0678.16008

GAP
Full Text:

### References:

 [1] Bhandari Ashwani K and Luthar Indar S, Certain conjugacy classes of units in integral group rings of metacyclic groups, J. Number Theory 18(2) (1984) 215–228 · Zbl 0527.16006 [2] Bovdi V, Höfert C and Kimmerle W, On the first Zassenhaus conjecture for integral group rings, Publ. Math. Debrecen 65(3–4) (2004) 291–303 · Zbl 1076.16028 [3] Bovdi V A, Jespers E and Konovalov A B, Integral group ring of the first Janko simple group, manuscript (eprint arXiv:math.GR/0608441) [4] Bovdi V A and Konovalov A B, Integral group ring of the first Mathieu simple group, Groups St. Andrews 2005, vol. 1, London Math. Soc. Lecture Note Ser., no. 340 (Cambridge: Cambridge Univ. Press) (2007) pp. 237–245 · Zbl 1120.16025 [5] Bovdi V A and Konovalov A B, Integral group ring of the Mathieu simple group M 23, Comm. Algebra, to appear (eprint arXiv:math.RA/0612640) · Zbl 1148.16027 [6] Bovdi V A, Konovalov A B and Siciliano S, Integral group ring of the Mathieu simple group M 12, Rend. Circ. Mat. Palermo (2) 56(1) (2007) 125–136 (eprint arXiv:math.RA/0612638) · Zbl 1125.16020 [7] Bovdi Victor and Hertweck Martin, Zassenhaus conjecture for central extensions of S 5, J. Group Theory, 11(1) (2008) 63–74 (eprint arXiv:math.RA/0609435) · Zbl 1143.16032 [8] Cohn James A and Livingstone Donald, On the structure of group algebras. I, Canad. J. Math. 17 (1965) 583–593 · Zbl 0132.27404 [9] Dokuchaev Michael A, Juriaans Stanley O and Milies César Polcino, Integral group rings of Frobenius groups and the conjectures of H. J. Zassenhaus, Comm. Algebra 25(7) (1997) 2311–2325 · Zbl 0881.16020 [10] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4 (2006) ( http://www.gap-system.org ) [11] Guralnick Robert M and Lorenz Martin, Orders of finite groups of matrices, Groups, Rings and Algebras, A conference in honor of Donald S Passman (eds) W Chin, J Osterburg and D Quinn, Contemp. Math., vol. 420 (Providence, RI: Amer. Math. Soc.) (2006) pp. 141–161 [12] Hertweck Martin, Torsion units in integral group rings of certain metabelian groups, Proc. Edinburgh Math. Soc. (2), to appear · Zbl 1151.16033 [13] Hertweck Martin, Partial augmentations and Brauer character values of torsion units in group rings, Comm. Algebra, to appear (eprint arXiv:math.RA/0612429v2) [14] Hertweck Martin, On the torsion units of some integral group rings, Algebra Colloq. 13(2) (2006) 329–348 · Zbl 1097.16009 [15] Höfert Christian and Kimmerle Wolfgang, On torsion units of integral group rings of groups of small order, Groups, rings and group rings, Lect. Notes Pure Appl. Math., vol. 248 (Boca Raton, FL: Chapman & Hall/CRC) (2006) pp. 243–252 · Zbl 1107.16031 [16] Luthar I S and Bhandari A K, Torsion units of integral group rings of metacyclic groups, J. Number Theory 17(2) (1983) 270–283 · Zbl 0527.16005 [17] Luthar I S and Passi I B S, Zassenhaus conjecture for A 5, Proc. Indian Acad. Sci. (Math. Sci.) 99(1) (1989) 1–5 · Zbl 0678.16008 [18] Luthar I S and Sehgal Poonam, Torsion units in integral group rings of some metacyclic groups, Res. Bull. Panjab Univ. Sci. 48(1–4) (1998) 137–153 [19] Luthar I S and Trama Poonam, Zassenhaus conjecture for certain integral group rings, J. Indian Math. Soc. (N.S.) 55(1–4) (1990) 199–212 · Zbl 0737.16020 [20] Luthar I S and Trama Poonam, Zassenhaus conjecture for S 5, Comm. Algebra 19(8) (1991) 2353–2362 · Zbl 0729.16021 [21] Marciniak Z, Ritter J, Sehgal S K and Weiss A, Torsion units in integral group rings of some metabelian groups. II, J. Number Theory 25(3) (1987) 340–352 · Zbl 0611.16007 [22] Salim Mohamed Ahmed M, Torsion units in the integral group ring of the alternating group of degree 6, Comm. Algebra 35(12) (2007) 4198–4204 · Zbl 1161.16023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.