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A generalized p-primary cotorsion abelian group G is a pa-injective, that is satisfies pa Ext (-, G) =0, iff Gt is pa-injective in the category of torsion abelian groups. Such a torsion group is generally pa-torsion complete, but an example shows that all its Ulm factors need not be complete. The injective properties of generally p-torsion complete groups are investigated. They are an injectively closed class, and the corresponding class of sequences is the class of pa-pure sequences with split completion when a is "accessible". Also, these groups are the pa-high injectives. This paper is motivated by the problem of determining the structure, say in the sense of [2], of the p-injective abelian groups, i.e. those groups G such that pa Ext (-, G)= 0. It is shown that a generalized p-primary cotorsion group G is pa-injective iff its torsion subgroup is pa-injective in the category of torsion abelian groups. Properties similar to those in [19] are shown for this latter concept, which we call "p1-torsion injective". An example is then given to show that unlike the general case a p02-torsion injective need not have complete groups as Ulm factors. The injective properties of the generallypa-torsion complete groups are investigated. It is shown that they form an injectively closed class and the corresponding sequences are characterized for accessible ordinals. Finally, the p'-high injectives are characterized. It is assumed that the reader is familiar with elementary homological algebra such as in [5] and [1, p. 116] and with ??1 and 2 of Chapter III in [14]. We follow Mines [15] in calling an ordinal accessible if it is the limit of a countable increasing sequence of lesser ordinals. The notation is that of [3] and/or [12]. 1. Background. Let G be an additively written abelian group. G is called a generalized p-primary group or g.p. group if qG = G for all primes q different from the given prime p. If a is a limit ordinal, we can make G a topological group by letting a fundamental neighborhood system of zero be {p0G}i<6a (see [15]). This induces a Hausdorff topology on GipaG. Completing this we get a group La,(G). We denote the torsion subgroup of La(G) by Ta(G). La(G) can also be viewed as La(G) = proj lim G/piG IJ

Generally 𝑝^{𝛼}-torsion complete abelian groups