Vol. 20, No. 1, 1967

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A representation theorem for abelian groups with no elements of infinite p-height

Delmar L. Boyer and Adolf G. Mader

Vol. 20 (1967), No. 1, 31–33

The purpose of this note is to give a generalization of the representation Theorems 33.1 and 33.2 of [2]. Let G be an arbitrary abelian group and B = [λΛxλ] [i1Bλ] be a p-basic subgroup of G, cf. [3], where λΛxλis the torsionfree part. For all λ Λ let (Fp)λ be a copy of the group of p-adic integers, and let (Fp)λ denote the infinite cyclic group of finite p-adic integers in (Fp)λ. Then G can be mapped homomorphically into the complete direct sum [λΛ(Fp)λ] [i1Bi] with kernel pωG. Furthermore, the image of G is a p-pure subgroup which contains [λΛ(Fp)λ] [i1Bi] as a p-basic subgroup and is in turn contained in the p-adic completion of this subgroup (See Section 1 for definitions). This representation is completely analogous to the representation theorem for p-groups which is contained as a special case, and hopefully it is of similar use.

Mathematical Subject Classification
Primary: 20.30
Received: 14 May 1965
Published: 1 January 1967
Delmar L. Boyer
Adolf G. Mader