  Correlated excitations in a plane circular configuration of identical atoms
with parallel dipole moments are investigated. The collective energy
eigenstates, their level shifts and decay rates are computed utilizing a
decomposition of the atomic state space into carrier spaces for the irreducible
representations of the symmetry group $\ZZ_N$ of the circle. It is shown that
the index $p$ of these representations can be used as a quantum number
analogously to the orbital angular momentum quantum number $l$ in hydrogen-like
systems. Just as the hydrogen s-states are the only electronic wave functions
which can occupy the central region of the Coulomb potential, the
quasi-particle corresponding to a collective excitation of the atoms in the
circle can occupy the central atom only for vanishing $\ZZ_N$ quantum number
$p$. If a central atom is present, the $p=0$ state splits into two and shows
level-crossing at certain radii; in the regions between these radii, damped
Rabi oscillations between two "extreme" $p=0$ configurations occur. The
physical mechanisms behind super- and subradiance at a given radius and the
divergence of the level shifts at small interatomic distances are discussed. It
is shown that, beyond a certain critical number of atoms in the circle, the
lifetime of the maximally subradiant state increases exponentially with the
number of atoms in the configuration, making the system a natural candidate for
a {\it single-photon trap}.
