A linear, electromagnetic analysis is presented of the cavity mode structure and dispersion relations for waves with frequency near the electron‐plasma frequency in cylindrical plasmas with monotonically decreasing electron density and finite magnetic field. Analytic solutions are obtained which are exact in the limits when either the ratio of electron cyclotron to plasma frequency or the ratio of plasma size to parallel wavelength tends to infinity. Comparison with numerical solutions confirms that the analytic results are highly accurate even for fairly modest values of these ratios. Thermal effects are incorporated, including Landau (damping or) gain, which show the lowest‐order transverse modes to have greatest gain. These modes are highly localized near the cylinder axis so that the plasma itself acts as a cavity, regardless of edge boundary conditions. The theory thus enables an interpretation to be made of maser action in quasicylindrical plasmas such as tokamaks.