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Let p be a prime number and let G be an abelian p{group. Let be the maximal normal p{subgroup of Aut G and the maximal p{subgroup of its centre. Let t be the torsion radical of E(G). Then = (1 + t) .T he result is new for p = 2 and 3, and the proof is new and valid for all primes p. x1. Introduction Throughout, p is an arbitrary but xed prime and G is an abelian p{group. We let E = E(G) denote the endomorphism ring of G ,A ut Gits automorphism group, and = Op(AutG) the maximal normal p{subgroup of AutG . The importance of in determining the structure of AutG, characterising the action of AutG on G and recovering invariants of G from AutG has long been recognized; see for example (3, Section 114), (6) and (1). For p 5 , considered as a group of units of E has been determined in a series of papers starting with Shoda (17) for nite groups, Freedman (2) for countable reduced groups, Hill (9) for totally projective groups and Hausen (5) for arbitrary p{groups: if t denotes the torsion radical of E (i.e. t is the ideal consisting of all elements of nite order in the Jacobson radical of E), then = 1 + t . Crucial for the proof is the following result, rst stated in (5). It was inspired by Freedman (2), stated by Leptin for k = 1 (11) and, in fact, is easily proven by induction on k following Leptin's arguments (11, p.101). The Pierce radical of E is dened to be P = f" 2E : p n G ( p ) " p n+1G for all integers n 0g: Clearly, P is a two{sided ideal in E containing the Jacobson radical of E (13, p. 289). Let 1 + P denote the coset containing 1 in the quotient ring E=P. The Freedman{Leptin Lemma. Let p 5 , let G be a reduced abelian p{ group, and let =1 + 2AutG \ (1 + P) . Then, for any natural number k , p k =1 if and only if pk =0 . For p 5 , the description of proceeds as follows: the ideal t is contained in the Jacobson radical of E ;s ince tconsists of all elements in P of nite order, it follows that 1 + t is a normal p{subgroup of AutG and thus is contained in (3, Theorem 114.4). Conversely, induces the identity mapping in all Ulm factors

The maximal normal 𝑝-subgroup of the automorphism group of an abelian 𝑝-group