Archiv der Mathematik

, Volume 113, Issue 1, pp 53–57 | Cite as

Burnside’s theorem in the setting of general fields

  • Heydar Radjavi
  • Bamdad R. YahaghiEmail author


We extend a well-known theorem of Burnside in the setting of general fields as follows: for a general field F the matrix algebra \(M_n(F)\) is the only algebra in \(M_n(F)\) which is spanned by an irreducible semigroup of triangularizable matrices. In other words, for a semigroup of triangularizable matrices with entries from a general field irreducibility is equivalent to absolute irreducibility. As a consequence of our result we prove a stronger version of a theorem of Janez Bernik.


Semigroup Quaternions Spectra Irreducibility Triangularizability 

Mathematics Subject Classification

15A30 20M20 


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This paper was submitted while the second named author was on leave of absence at the University of Waterloo. He would like to thank Department of Pure Mathematics of the University of Waterloo, and in particular Professors L.W. Marcoux and H. Radjavi, for their support.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Mathematics, Faculty of SciencesGolestan UniversityGorganIran

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