Catalyzes the cleavage of glutathione into 5-oxo-L-proline and a Cys-Gly dipeptide. Acts specifically on glutathione, but not on other gamma-glutamyl peptides. glutathione = 5-oxo-L-proline + L-cysteinylglycine Belongs to the gamma-glutamylcyclotransferase family. ChaC subfamily.