Burnside’s theorem in the setting of general fields
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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.
KeywordsSemigroup Quaternions Spectra Irreducibility Triangularizability
Mathematics Subject Classification15A30 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.
- 9.Yahaghi, B.R.: Burnside type theorems in real and quaternion settings. arXiv:1710.03849v2