Let p be an odd prime. Let \({G = SL_2(\mathbb {F}_p)}\) and let B denote the subgroup of upper triangular matrices of G. Finally, let \({\mathbb {F}}\) be an algebraically closed field of characteristic p. The Green correspondence gives a bijection between the non-projective indecomposable \({\mathbb {F}[G]}\) modules and non-projective indecomposable \({\mathbb {F}[B]}\) modules, realised by restriction and induction. In this paper, after recalling a suitable description of the non-projective indecomposable modules for these group algebras, we explicitly describe the Green correspondence bijection. We do this by pinpointing the modules’ position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of this description: formula for lifting the \({\mathbb {F}[B]}\) module decomposition of an \({\mathbb {F}[G]}\) module, and a complete description of \({\text { Ind}_B^G}\) and \({\text { Res}^G_B}\) .
