Subnormal subgroups in division rings with generalized power central group identities

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• 作者：Mai Hoang Bien
• 关键词：Division ring ; Subnormal subgroup ; Power central ; Generalized group identity
• 刊名：Archiv der Mathematik
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：106
• 期：4
• 页码：315-321
• 全文大小：438 KB
• 参考文献：1.Bien M. H.: On some subgroups of \({D^*}\) which satisfy a generalized group identity. Bull. Korean Math. Soc. 52 (2015), 1353–1363.MathSciNet CrossRef MATH
  2.Chebotar M. A., Fong Y., Lee P. -H.: On division rings with algebraic commutators of bounded degree. Manuscripta Math. 113 (2004), 153–164.MathSciNet CrossRef MATH
  3.Chiba K.: Skew fields with a nontrivial generalised power central rational identity. Bull. Austral. Math. Soc. 49 (1994), 85–90.MathSciNet CrossRef MATH
  4.Golubchik I. Z., Mikhalev A. V.: Generalized group identities in the classical groups. Zap. Nauch. Sem. Leningrad. Otdel. Inst. Steklov. 114 (1982), 96–119.MathSciNet MATH
  5.Goncalves J. Z., Mandel A.: Are there free groups in division rings?. Israel J. Math. 53 (1986), 69–80.MathSciNet CrossRef MATH
  6.B. X. Hai, M. H. Bien, and T. T. Deo, On Gelfand-Kirillov dimension of weakly locally finite division rings, Preprint.
  7.Hai B. X., Deo T. T., Bien M. H.: On subgroups in division rings of type 2. Studia Sci. Math. Hungar. 49 (2012), 549–557.MathSciNet MATH
  8.Herstein I. N.: Multiplicative commutators in division rings. II. Rend. Circ. Mat. Palermo 29 (1980), 485–489.MathSciNet CrossRef MATH
  9.Herstein I. N.: Multiplicative commutators in division rings. Israel J. Math. 31 (1978), 180–188.MathSciNet CrossRef MATH
  10.Herstein I. N., Procesi C., Schacher M.: Algebraic valued functions on noncommutative rings. J. Algebra 36 (1975), 128–150.MathSciNet CrossRef MATH
  11.Mahdavi-Hezavehi M.: Commutators in division rings revisited. Bull. Iranian Math. Soc. 26 (2000), 7–88.MathSciNet MATH
  12.Mahdavi-Hezavehi M., Akbari-Feyzaabaadi S., Mehraabaadi M., Hajie-Abolhassan H.: On derived groups of division rings. II. Comm. Algebra 23 (1995), 2881–2887. MathSciNet CrossRef MATH
  13.Tomanov G. M.: Generalized group identities in linear groups. Math. USSR, Sbornik 51 (1985), 33–46.CrossRef MATH
  14.Zhang J. J.: On Gelfand-Kirillov transcendence degree. Trans. Amer. Math. Soc. 348 (1996), 2876–2899.MathSciNet MATH
  
• 作者单位：Mai Hoang Bien (1)
  
  1. Department of Basic Sciences, University of Architecture, 196 Pasteur Str., Dist. 1, Ho Chi Minh City, Vietnam
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8938

文摘

Let \({D}\) be a division ring with center \({F}\). The aim of the paper is to show that if \({F}\) is uncountable or \({D}\) is finite dimensional over \({F}\), then every subnormal subgroup of the multiplicative group \({D^*}\) of \({D}\) satisfying a nontrivial generalized power central group identity is contained in \({F}\). As a corollary, Conjecture 2 in (Herstein, Israel J Math 31:180–188, 1978) holds in case \({D}\) is finite dimensional.