Electromagnetically induced transparency (EIT) is an intriguing phenomenon in quantum optics, and has a wide range of applications in the fields of quantum information processing, quantum precision metrology, etc. Recently, with the rapid development of the generation and detection of vortex beams, the combination of EIT and vortex beams offers new and novel functionalities and applications, such as EIT-based atomic compass, spatially dependent EIT, and storage, transfer, and arithmetic of orbit angular momentum of light. For these investigations and applications, the propagation and evolution of vortex beams in EIT media are of crucial importance. In this article, the propagation evolution properties of a Laguerre-Gaussian (LG) vortex beam in an EIT medium with Λ-type three-level atoms are studied. Based on the generalized Huygens-Fresnel principle and the ABCD transfer matrix of the EIT medium, the analytical expression of the LG vortex beam propagatingin the EIT medium is derived. The propagation evolution of the intensity and phase of the beam and their dependences on the parameters such as the topological charge, the radial index, and the coherence length are studied. It is shown that the LG vortex beam propagating in the EIT medium experiences focusing and diverging periodically. For a partially coherent LG vortex beam, a smaller topological charge or larger radial index leads to the hiding of the phase singularity. Besides, the phase singularity splits and the phase singularities with opposite topological charges are generated during propagation. As the increasing of the coherence length of the beam, the hiding, splitting and generating of the phase singularities vanish. The results could find applications in developing new EIT-based techniques.