  We study the Lie algebra structure of the first Hochschild cohomology group
of a finite dimensional monomial algebra A, in terms of the combinatorics of
its quiver, in any characteristic. This allows us also to examine the identity
component of the algebraic group of outer automorphisms of A in characteristic
zero. Criteria for the (semi-)simplicity, the solvability, the reductivity, the
commutativity and the nilpotency are given.
