Abstract: Let k be a commutative ring, G be a group, and kG be the group algebra. We mainly study the fibrant dimension of modules over the group algebra kG. First, we establish the relationships between the fibrant dimension of modules and injective dimension, Gorenstein injective dimension. Then, we give some equivalent characterizations of the fibrant dimension of modules. Finally, we discuss the existence of fibrant preenvelopes of modules and their relationship with Gorenstein injective preenvelopes.