Catalyzes the transfer of a phosphate group to glutamate to form L-glutamate 5-phosphate. ATP + L-glutamate = ADP + L-glutamyl 5-phosphate Amino-acid biosynthesis; L-proline biosynthesis; L-glutamate 5-semialdehyde from L-glutamate: step 1/2. Belongs to the glutamate 5-kinase family.