  The algebra S of symmetric invariants over the field with two elements is an
unstable algebra over the Steenrod algebra A, and is isomorphic to the mod two
cohomology of BO, the classifying space for vector bundles. We provide a
minimal presentation for S in the category of unstable A-algebras, i.e.,
minimal generators and minimal relations.
  From this we produce minimal presentations for various unstable A-algebras
associated with the cohomology of related spaces, such as the BO(2^m-1) that
classify finite dimensional vector bundles, and the connected covers of BO. The
presentations then show that certain of these unstable A-algebras coalesce to
produce the Dickson algebras of general linear group invariants, and we
speculate about possible related topological realizability.
  Our methods also produce a related simple minimal A-module presentation of
the cohomology of infinite dimensional real projective space, with filtered
quotients the unstable modules F(2^p-1)/A bar{A}_{p-2}, as described in an
independent appendix.
