To suppress the secondary electron multipactor on dielectric surfaces of a dielectric load accelerator under an electromagnetic field in TM mode, the method of adopting both groove structure and external axial magnetic field is introduced. As the electric field distribution of the TM mode is composed of both normal and tangential components, it is different from that under the condition of dielectric window in HPM. Thus, theoretical analysis and numerical simulation are employed to study the movement of electrons under different conditions: such as dielectric surface shapes, electric field strength, and magnetic field strength etc. Based on the particle-in-cell (PIC) simulation, the collision energy and transmit-duration of secondary electrons in different groove structures and axial magnetic fields are compared with one another. Results show that the magnetic field is useful for suppressing the development of secondary electron on dielectric surface, while it is not very efficient under high electric field strength. The method of introducing groove structure and certain axial magnetic field on dielectric surface at the same time is capable of affecting the movement of electrons in electric field of different strength. So it is great helpful in improving the ability of multipactor suppression, which is significant for improving the threshold of breakdown on dielectric surface and the power of cavity. However, a too high or too low magnetic field is not very useful for the suppression of multipactor. Furthermore, employing only one of the two parts of the method is also less effective in suppressing the multipactor.