A cap in a projective space is a set of points no three of which are collinear. A cap is called complete if it is maximal subject to set-theoretical inclusion. Let \(G\) be a cyclic Singer group of the \(n\)-dimensional projective space \(PG( n,q)\). Let \(H\) be a subgroup of \(G\) and put \(|H|=N\). In this paper, the author studies the following problem: when is a point orbit of \(H\) a cap? A necessary and sufficient condition for this is derived and used to give a short proof of some results by \textit{E. Ebert} [Can. J. Math. 37, 1163-1175 (1985; Zbl 0577.51001)] and to show that small orbits are typically caps.