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Subject and Purpose. The effects considered concern phase synchronization of electrons in an ideal plasma subjected to the action of an external uniform, d. c. magnetic field. Two modes of the synchronization are discussed, specifically one by an external electromagnetic field and the other by the cyclotron radiation emitted by the electrons. The purpose is to compare these forms of synchronization and their effects on plasma stability. Methods and Methodology. The plasma is represented as a set of coupled oscillators whose dynamics is described via coupled differential equations. Assuming the coupling between the oscillators to be weak we find analytical solutions to the equation, further performing a stability analysis which exploits standard approaches of the dynamical systems theory. The solutions found are validated through corresponding numerical simulations. Results. As has been found, an external electromagnetic wave may be capable of guiding the particles toward phase synchronization, which can lead to formation of phased bunches. This mechanism of particle grouping may prove to be more efficient, in terms of scale times of synchronization, if compared with known mechanisms exploiting relativistic effects. Additionally, we show that the cyclotron radiation emitted by the charged particles (which is often disregarded because of its smallness) can lead to self-phase synchronization of the electrons. Moreover, should the density of charged particles in the ensemble be sufficiently high, an instability can arise, potentially disrupting the ensemble. Estimates have been provided of the level of random fluctuations capable of undermining the synchronization process and plasma dynamics stabilization. Conclusions. The most significant finding of this analysis is the emergence of low-frequency oscillations in the charged oscillators set, followed by an onset of the plasma instability when the plasma density exceeds a certain critical value. Within that scenario, the ensemble of oscillators sitting in the external magnetic field is no longer held together by the field. The effect should be taken into account in applications related to plasmas of a relatively high density. Key words: cyclotron radiation; cyclotron resonance; particle dynamics; plasma, synchronization