Overgroups of exterior powers of an elementary group. Normalizers

We establish two characterizations of an algebraic group scheme {{2pt}{ᵐ}}GLₙ over Z . Geometrically, the scheme {{2pt}{ᵐ}}GLₙ is a stabilizer of an explicitly given invariant form or, generally, an invariant ideal of forms. Algebraically, {{2pt}{ᵐ}}GLₙ is isomorphic (as a scheme over Z ) to a normalizer of the elementary subgroup functor {{2pt}{ᵐ}}Eₙ and a normalizer of the subscheme {{2pt}{ᵐ}}SLₙ .Our immediate goal is to apply both descriptions in the “sandwich classification” of overgroups of the elementary subgroup. Additionally, the results can be seen as a solution of the linear preserver problem for algebraic group schemes over Z , providing a more functorial description that goes beyond geometry of the classical case over fields.