A short survey of Burnside type theorems
by
Bamdad R. Yahaghi
School of Mathematics, IPM, Tehran, Iran

A version of a celebrated theorem of Burnside asserts that M_n(F) is the only irreducible subalgebra of M_n(F) provided that the field F is algebraically closed. In other words, Burnside's theorem characterizes all irreducible subalgebras of M_n(F) whenever F is algebraically closed. In view of this, by a Burnside type theorem for certain irreducible subalgebras of matrices, we mean a result which characterizes such subalgebras. In this talk, we present a simple proof of Burnside's theorem. We also present Burnside type theorems for irreducible subalgebras of M_n(R), a result which is well known to the experts, and for irreducible subalgebras of M_n(H), where H denotes the division ring of quaternions. For a given n > 1, we characterize all fields F for which Burnside's Theorem holds in M_n(F). If time permits, letting K be a field and F a subfield of K which is k-closed for all k dividing n with k > 1, we present a Burnside type theorem for irreducible F-algebras of matrices in M_n(K) on which trace is not identically zero. (For a k > 1, a field F is said to be k-closed if every polynomial of degree k over F is reducible over F, e.g., R is k-closed for any odd integer greater than one.)

Date received: April 7, 2008