Primitive arcs in PG(2, q)

We show that a complete arc K in the projective plane PG(2, q) admitting a transitive primitive group of projective transformations is either a cyclic arc of prime order or a known arc. If the completeness assumption is dropped, then K has either an affine primitive group, or K is contained in an explicit list. In order to find these primitive arcs, it is necessary to determine all complete k-arcs fixed by a projective elementary abelian group of order k. As a corollary to our result, we list all complete arcs fixed by a 2-transitive projective group.