We present the first measurements of inter-subband absorption of microwaves in surface electrons on liquid helium subjected to perpendicular magnetic field B. In quantizing B, the power absorption shows intermittent regions of enhanced and suppressed absorption. This behavior is caused by strong variation of the excited-electron decay rate with B. Particularly, fast decay due to elastic scattering provides condition for strong absorption and overheating of the electron system, while slow decay due to inelastic scattering limits absorption and causes its saturation. An unexpected feature is the strong suppression of absorption at magnetic fields where the inter-subband energy splitting is a multiple number of the cyclotron energy. PACS numbers: 67.90.+z, 73.20.At, 78.70.Gq Surface state electrons (SSEs) on liquid helium provide a unique example of a two-dimensional electron system 1 . Below 1 K, they are suspended in a vacuum about 10 nm above the liquid because of a strong repulsive barrier at the free surface. An attractive image potential is responsible for a subband structure with a discrete 1D energy spectrum ǫ n = R/n 2 , where n = 1, 2, .. and R = 7.6 (4.2) K for liquid 4 He ( 3 He). Such electrons are strongly-correlated 2-5 , very mobile, 6,7 and interact very weakly with the environment mostly through coupling to surface capillary waves (ripplons). Great interest in SSEs comes from several proposals to use electrons interacting with resonant microwave radiation for quantum computing as this system promises a scalable architecture for qubits with a very long coherence time [8][9][10][11][12][13][14] . In particular, these proposals stimulated recent studies of the inter-subband microwave absorption in SSEs 15-17 . Collin et al. found that temperature-dependent contribution to the absorption linewidth γ is small at low T and obtained a relatively large value of Ω R /γ ≈ 300 (Ω R is the Rabi frequency), which estimates the number of qubit operations per coherence time 15 . The Rabi frequency was estimated from the observed saturation of absorption assuming that electrons populate only the two lowest surface states. Later, it was shown that the excited-electron decay due to elastic scattering from the vapor atoms or ripplons provide an efficient channel to transfer the excitation energy into the kinetic energy of electron in-plane motion 17 . This causes strong overheating of the electron system, which results in population of higher excited states and absorption bleaching. The latter produces features similar to the absorption saturation in a two-level system but can be adequately described only taking into account the contribution from many energy levels 17 .The situation is different in a quantizing magnetic field B applied perpendicular to the surface, which confines the in-plane states of electrons to Landau levels E l =hω c (l − 1/2), l = 1, 2, .., where ω c = eB/m is the cyclotron frequency. Then, the fast decay of the microwave-excited electron due to elastic scattering can be suppressed providing...