We investigate the Compton scattering of a photon by an electron residing i
n the ground Landau level in a strong external magnetic field under both re
sonant and nonresonant conditions. It is demonstrated that the cross sectio
n of this process displays an oscillatory behavior as a function of the fre
quency of an incident photon. For the fields of similar to 10(12) G, the cy
clotron-resonance cross section is six orders of magnitude higher than the
nonresonant cross section.