Linear Transformations That Preserve The Nilpotent Matrices

Let sln be the algebra of n × n matrices with zero trace and entries in a field with at least n elements. Let N be the set of nilpotent matrices. The main result in this paper is that the group of nonsingular linear transformations L on sln such that L(N) = N is generated by the inner automorphisms: X → S−1XS; the maps: X → aX, for a≠0; and the map: X → Xt that sends a matrix X to its transpose.