Catalyzes the transfer of a phosphate group to glutamate to form glutamate 5-phosphate which rapidly cyclizes to 5-oxoproline. ATP + L-glutamate = ADP + L-glutamyl 5-phosphate Amino-acid biosynthesis; L-proline biosynthesis; L-glutamate 5-semialdehyde from L-glutamate: step 1/2. Present with 10400 molecules/cell in log phase SD medium. Belongs to the glutamate 5-kinase family.