  The point-pushing subgroup P(S) of the mapping class group Mod(S) of a
surface with marked point is an embedding of \pi_1(S) given by pushing the
marked point around loops. We prove that for g>=3, the subgroup P(S) is the
unique normal, genus g surface subgroup of Mod(S). As a corollary to this
uniqueness result, we give a new proof that Out(Mod^{\pm}(S))=1, where Out
denotes the outer automorphism group; a proof which does not use automorphisms
of complexes of curves. Ingredients in our proof of this characterization
theorem include combinatorial group theory, representation theory, the Johnson
theory of the Torelli group, surface topology, and the theory of Lie algebras.
