A combination of analytical calculations and Monte Carlo simulations is used to find the ground-state structures in monodisperse ferrofluid monolayers under the influence of an external magnetic field. We study two different regimes, where (i) the direction of an external field is perpendicular to the monolayer plane and (ii) an external magnetic field is in the plane. In the field perpendicular to the plane we observe a transition from an ideal ferroparticle ring to a hexagonal structure. The analytical value of the critical field strength needed for this transition is obtained and shown for relatively large systems to be independent of the number of particles. For smaller systems the value of the critical field is system-size dependent and grows fast to its asymptotic value with increasing number of particles. In the case where the magnetic field is aligned parallel to the layer plane, the critical field needed to break a ring and to create a new ground-state structure, namely a ferroparticle chain, is much smaller than in the case where a field is applied perpendicularly to the plane. The analytical expression for the asymptotic critical field in case of in-plane field is found to be a decreasing function of the total particle number in a system. We characterize both transitions to be of the first-order-type transition, with magnetic susceptibility diverging near the critical point. Our studies show that the domination of the in-plane correlations of dipoles within the ferroparticle ring results in much lower magnetic in-plane susceptibility in comparison to the one perpendicular to the plane.