B.C. Kestenband , J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas , E. Boros, and T. Szönyi  constructed complete (q2 − q + l)-arcs in PG(2, q2), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q2 − q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.