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Let G be a ^-primary abelian group, that is, let G be a commutative group each element of which has finite order equal to a power of a fixed prime p. We suppose that G is written additively and we say that an element g E G is divisible by a positive integer n if there is a solution in G to the equation nx = g. Note that 0 is divisible by every positive integer and that the collection of all the elements of G with this property is a subgroup of G. This subgroup is denoted by pG. I t is immediate that £(G/£G) = 0. We can define pG for any ordinal a inductively by taking intersections at limit ordinals and letting p«+iG=p(pG). I t has been known for a long time that if the primary group G is countable, then any automorphism of pG can be extended to an automorphism of G; this result is implicitly contained in Zippin's proof [7] of Ulm's theorem. I t is worth noting that if G is countable, then G/pG is a direct sum of cyclic groups according to a classical result due to Prüfer [6 ] : if G is a countable primary group such that p<*G = 0, then G must be a direct sum of cyclic groups. This suggests the possibility of extending Zippin's result to the case where G is any primary group with the property that G/p^G is a direct sum of cyclic groups. Recently, this result has been obtained by Hill and Megibben [3], [4], in a more general setting, and independently by Crawley [l ]. What seems to the author as being a more striking generalization is the result being announced in this note, which says that the subgroup does not have to be pG in order to extend height-preserving automorphisms. By the height, ho{x), of an element x in G we mean the largest ordinal a such that xÇzpG if such a largest ordinal a exists. Since pG = na<p p G when fl is a limit, the only failure of the existence of such a largest ordinal is when x(EpG for all a. In that case, we put ho(x) = 00 and adopt the convention that 00 > ce for all ordinals a. If 7T is an automorphism of a subgroup H of G, we say that 7r is heightpreserving in G if ho(x)=hG(T(x)) for all x in H. Denote by d°(H) the group of automorphisms of H that preserve heights in G and let ®>H(G) denote the group of automorphisms of G that map H onto H. Naturally, we write a(G) for &(G) since the latter is the full automorphism group of G, and we write Ofc(G) instead of Cto(G) for the

Extending automorphisms on primary groups