Abstract We propose monolayer graphene system and analyze the effects of giant Kerr nonlinearity and Doppler broadening. The focus is to construct and design experimentally viable model by which we can enhance the group index of the medium so that we have slow group velocity. For obtaining the desired goal, we consider Landau level of graphene monolayer placed perpendicular to magnetic field M Tokman et al (2013, Phys. Rev. lett. 110, 077404). Giant Kerr nonlinearity and Doppler broadening enhance the medium group index greatly, resulting in a slow group velocity that has uses in storage devices, optical buffering, quantum memory, and data processing. We also investigate the behavior exhibited by the weak probe’s transmission spectrum, which can change when the terahertz signal frequency is turned on and off. Advantages of under consideration model are its a 2D material whereas other materials are semiconductor or gaseous state. Also, Landau level of graphene has domains tuneable in tera hertz and infra-red region while we cannot tune the semiconductor and atomic media once the transition is chosen. Therefore, the proposed magnetized graphene system offers a tremendous opportunity for incorporating quantum interference into the development of optical solid-state devices.