On a relative version of a theorem of Bogomolov over perfect fields and its applications

In this paper, we investigate some aspects of representation theory of reductive groups over non-algebraically closed fields. Namely, we state and prove relative versions of a well-known theorem of Bogomolov and derive from it as consequence, a relative version of a theorem of Sukhanov, which are related to observable subgroups of linear algebraic groups over non-algebraically closed perfect fields.