Let Comp be the category of compact spaces and continuous mappings. A functor F: Comp\(\to Comp\) is called normal if F is continuous and preserves weights, monomorphisms, epimorphisms, intersections, preimages, singletons and the empty set. It is proved that a normal functor F is the ith power functor for some natural number i whenever F preserves either finite products or absolute retracts. If CG is the category of compact separable topological groups and continuous homomorphisms with the natural forgetful functor U: CG\(\to Comp\) then for a normal functor F: Comp\(\to Comp\) there exists a functor G: CG\(\to CG\) with \(U\circ G=F\circ U\) if and only if F is the ith power functor for some natural number i.