  The electron mass operator in a strong magnetic field is calculated. The
contribution of higher Landau levels of virtual electrons, along with the
ground Landau level, is shown to be essential in the leading log approximation.
The effect of the electron dynamical mass generation by a magnetic field is
investigated. In a model with N charged fermions, it is shown that some
critical number N_{cr} exists for any value of the electromagnetic coupling
constant alpha, such that the fermion dynamical mass is generated with a
doublet splitting for N < N_{cr}, and the dynamical mass does not arise at all
for N > N_{cr}, thus leaving the chiral symmetry unbroken.
