Let k be an algebraically closed field of positive characteristic p, x a proper integral k-scheme o f finite type and G b e a commutative affine k-group scheme. For a k-prescheme T , the isomorphism classes o f principal fibre spaces Y over X T with group G form an abelian group with the well-known multiplication. We shall denote this abelian group by PH(G, X / k ) ( T ) . Then the functor X / k)(T ) is a contravariant functor from the category o f k-preschemes (Sch/k) to the category o f abelian groups (Ab). The associated sheaf o f PH(G, X / k ) with respect to th e (fpqc)topology of (Sch/k) is denoted by PH(G, X/k). If G is the multiplicative group Gm ., PH(G,„, X/ k ) coincides with the Picard functor Pic(X/k) o f X, and P ic (X / k ) is representable by a commutative k-group scheme, locally of finite type over k. The purpose o f this paper is to study the representability of the functor PH(G, X /k ) for an arbitrary commutative affine k-group scheme of finite type. If G is the additive group G „, PH(G„ X/k) is representable by L ie(P ic(X / k)) which is isomorphic to a direct product of G . If G is a simple finite k-group scheme (i.e. G=trp, P p > (Z/ PZ)k and (Z / qZ ),; q : prime, (p , q )= 1 ), P H (G , X / k ) is * ) This article was presented as a d octoral thesis to the Faculty of Science.