We establish two characterizations of an algebraic group scheme GLn over Z. Geo-metrically, the schemeVm GLn is a stabilizer of an explicitly given invariant form or, generally, an invariant ideal of forms. Algebraically,Vm GLn is isomorphic (as a scheme over Z) to a normalizer of the elementary subgroup functorVm En and a normalizer of the subschemeVm 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. © 2024 Deutsche Mathematiker-Vereinigung.