The Equivalence of High Subgroups

Two subgroups of a group G are called equivalent if there is an automorphism of G that maps one of the subgroups onto the other. Suppose that G is a /7-primary abelian group and that A is an ordinal. A subgroup H oî G is //-high in G if H is maximal in G with respect to having zero intersection with p G. Under certain conditions on the quotient group G/pxG slightly weaker than total projectivity, it is shown, for a given \, that any two//-high subgroups of G are equivalent. In particular, if G/p"G is p"+ '-projective, the p^-high subgroups of G are all equivalent. D. Cutler [1] has recently proved the following result, where fa(G) denotes the ath Ulm invariant of G. Theorem. Let G be an abelian p-group such that G/p"G is p"+"-projective, and if n > 1 then fa+m(G) = 0 for 0 < m < n — 1. Then any two high subgroups of G are isomorphic. Recall that H is called a high subgroup of the abelian /»-group G if H is maximal with respect to having zero intersection with pwG. More generally, if H is maximal with respect to having zero intersection with pxG, then H is said to be //-high in G. Notation and terminology generally are in agreement with [2]. The case n = 1 in Cutler's theorem avoids certain technicalities, gains simplicity, and is of special interest. Corollary. Let G be an abelian p-group such that G/puG is p"+x-projective. Then any two high subgroups of G are isomorphic. If G/p^G satisfies the stronger hypothesis of being /»"-projective, which is equivalent to being a direct sum of cyclic groups, the preceding corollary is contained in the pioneer work on high subgroups by J. Irwin and E. Walker [7], who first raised and developed interest in the question: for which abelian /»-groups G are all of its high subgroups isomorphic? As Cutler mentioned in his paper, the first example of a group G with nonisomorphic high subgroups was constructed in [3]. However, it was soon apparent through the joint work of the author and C. Megibben [6] that the case of possessing nonisomorphic high subgroups is the rule rather than the Received by the editors August 1, 1982. 1980 Mathematics Subject Classification. Primary 20K10, 20K27.