Let p be a prime number and let G be an abelian p-group. Let Delta be the maximal normal p-subgroup of Aut G and zeta the maximal p-subgroup of its centre. Let t be the torsion radical of epsilon(G). Then Delta = (1 + t)zeta. The result is new for p = 2 and 3, and the proof is new and valid for all primes p.