  We study dynamical symmetry breaking on a cylinder in external magnetic field
parallel to the axis of cylinder when magnetic field affects the dynamics of
fermions only through the Aharonov-Bohm phase. We find that unlike other
previously studied cases magnetic field in our case counteracts the generation
of dynamical fermion mass which decreases with magnetic field. There exists
also a purely kinematical contribution to the fermion gap which grows linearly
with magnetic field. Remarkably, we find that the total fermion gap, which
includes both the dynamical and kinematical contributions, always increases
with magnetic field irrespectively the values of coupling constant and the
radius of cylinder. Thus, although the dynamical mass is suppressed, external
magnetic field does enhance the total fermion gap in the spectrum.
