LET G BE A FINITE P-GROUP, WHERE P IS A PRIME NUMBER AND AUT(G) BE THE AUTOmorphism GROUP OF G. WE PROVE THAT IF AUT(G) IS ISOMORPHIC TO GL(N, P) FOR SOME POSITIVE INTEGER N, THEN G IS AN ELEMENTARY ABELIAN GROUP OF ORDER PN. MOREOVER WE PROVE THAT IF G IS A FINITE ABELIAN P-GROUPS WITH |AUT(G)| = |GL(D, P)|, THEN G IS AN ELEMENTARY ABELIAN GROUP OF ORDER PD, WHERE D IS THE MINIMAL NUMBER OF GENERATORS OF G.