Burnsideʹs theorem: irreducible pairs of transformations Original Research Article

By Burnsideʹs theorem, if the linear transformations A and B, acting on a finite-dimensional complex vector space image, have no common nontrivial invariant subspaces, the words in A and B span image. Call the minimum spanning length of the pair {A,B} the smallest positive integer l with the property that words in A and B of length at most l span image. Let msl(A,B) denote the minimum spanning length. If image, msl(A,B)=2 and if image, msl(A,B)=3 or 4. If image, msl(A,B)less-than-or-equals, slantn2−3. If image then (i) msl(A,B)=2n−2 if {A,B,AB,BA} is linearly dependent, (ii) if B is unicellular, then msl(A,B)less-than-or-equals, slant2n−2, where the inequality is sharp, and it can happen that msl(A,B)=n.