Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 855-865, 2017


On the derived length of units in group algebra

Dishari Chaudhuri, Anupam Saikia

Received April 27, 2016.   First published August 10, 2017

Abstract:  Let $G$ be a finite group $G$, $K$ a field of characteristic $p\geq17$ and let $U$ be the group of units in $KG$. We show that if the derived length of $U$ does not exceed $4$, then $G$ must be abelian.
Keywords:  group algebra; group of units; derived subgroup
Classification MSC:  16S34, 16U60
DOI:  10.21136/CMJ.2017.0205-16


References:
[1] C. Bagiński: A note on the derived length of the unit group of a modular group algebra. Commun. Algebra 30 (2002), 4905-4913. DOI 10.1081/AGB-120014675 | MR 1940471 | Zbl 1017.16022
  [2] Z. Balogh, Y. Li: On the derived length of the group of units of a group algebra. J. Algebra Appl. 6 (2007), 991-999. DOI 10.1142/S0219498807002624 | MR 2376796 | Zbl 1168.16016
  [3] J. M. Bateman: On the solvability of unit groups of group algebras. Trans. Amer. Math. Soc. 157 (1971), 73-86. DOI 10.2307/1995832 | MR 0276371 | Zbl 0218.20006
  [4] A. Bovdi: The group of units of a group algebra of characteristic $p$. Publ. Math. 52 (1998), 193-244. MR 1603359 | Zbl 0906.16016
  [5] A. Bovdi: Group algebras with a solvable group of units. Commun. Algebra 33 (2005), 3725-3738. DOI 10.1080/00927870500243213 | MR 2175462 | Zbl 1082.16036
  [6] A. A. Bovdi, I. Khripta: Finite dimensional group algebras having solvable unit groups. Trans. Science Conf. Uzhgorod State University (1974), 227-233.
  [7] A. A. Bovdi, I. I. Khripta: Group algebras of periodic groups of a solvable multiplicative group. Math. Notes 22 (1977), 725-731. (In English. Russian original.); translation from Mat. Zametki 22 (1977), 421-432. DOI 10.1007/BF02412503 | MR 0485969 | Zbl 0363.16004
  [8] F. Catino, E. Spinelli: On the derived length of the unit group of a group algebra. J. Group Theory 13 (2010), 577-588. DOI 10.1515/JGT.2010.008 | MR 2661658 | Zbl 1205.16030
  [9] H. Chandra, M. Sahai: Group algebras with unit groups of derived length three. J. Algebra Appl. 9 (2010), 305-314. DOI 10.1142/S0219498810003938 | MR 2646666 | Zbl 1209.16028
  [10] H. Chandra, M. Sahai: On group algebras with unit groups of derived length three in characteristic three. Publ. Math. 82 (2013), 697-708. DOI 10.5486/PMD.2013.5461 | MR 3066439 | Zbl 1274.16046
  [11] D. Chaudhuri, A. Saikia: On group algebras with unit groups of derived length at most four. Publ. Math. 86 (2015), 39-48. DOI 10.5486/PMD.2015.6012 | MR 3300576 | Zbl 1347.16021
  [12] D. Gorenstein: Finite Groups. Chelsea Publishing Company, New York (1980). MR 0569209 | Zbl 0463.20012
  [13] J. Kurdics: On group algebras with metabelian unit groups. Period. Math. Hung. 32 (1996), 57-64. DOI 10.1007/BF01879732 | MR 1407909 | Zbl 0857.20001
  [14] G. T. Lee, S. K. Sehgal, E. Spinelli: Group rings with solvable unit groups of minimal derived length. Algebr. Represent. Theory 17 (2014), 1597-1601. DOI 10.1007/s10468-013-9461-8 | MR 3260911 | Zbl 1309.16025
  [15] K. Motose, Y. Ninomiya: On the solvability of unit groups of group rings. Math. J. Okayama Univ. 15 (1972), 209-214. MR 0322033 | Zbl 0255.20006
  [16] K. Motose, H. Tominaga: Group rings with solvable unit groups. Math. J. Okayama Univ. 15 (1971), 37-40. MR 0306297 | Zbl 0253.16012
  [17] D. S. Passman: Observations on group rings. Commun. Algebra 5 (1977), 1119-1162. DOI 10.1080/00927877708822213 | MR 0457540 | Zbl 0366.16003
  [18] M. Sahai: Group algebras with centrally metabelian unit groups. Publ. Mat., Barc. 40 (1996), 443-456. DOI 10.5565/PUBLMAT_40296_14 | MR 1425630 | Zbl 0869.16024
  [19] M. Sahai: On group algebras $KG$ with $U(KG)'$ nilpotent of class at most 2. Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications Int. Conf., Chennai 2006, Contemporary Mathematics 456, American Mathematical Society, Providence S. K. Jain (2008), 165-173. DOI 10.1090/conm/456/08889 | MR 2416149 | Zbl 1158.16017
  [20] A. Shalev: Meta-abelian unit groups of group algebras are usually abelian. J. Pure Appl. Algebra 72 (1991), 295-302. DOI 10.1016/0022-4049(91)90067-C | MR 1120695 | Zbl 0735.16020
  [21] W. S. Yoo: The structure of the radical of the non semisimple group rings. Korean J. Math. 18 (2010), 97-103.

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Affiliations:   Dishari Chaudhuri, Anupam Saikia, Department of Mathematics, Indian Institute of Technology Guwahati, Near Doul Gobinda Road, Amingaon, Pin-781039, Guwahati, Assam, India, e-mail: dishari@iitg.ernet.in, a.saikia@iitg.ernet.in

 
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