On isotype subgroups of abelian groups
摘要
In his book Abelian groups^ L. Focus asks the following question. Let G be a jo-group and H be a subgroup without elements of infinite height. Lnder what conditions can H be embedded in a pure subgroup of the same power and again without elements of infinite height? {See [2], p. 96.) This question has been answered by Charles [1] and IRWIN [3]. Irwin's solution was effected by showing that any subgroup maximal with respect to disjointness from the subgroup of elements of infinite height is pure. For /^-groups, the subgroups of element of infinite height is p^G. Now for any Abelian group G^ any prime/?, and any ordinal a, one may define p^G^ and this suggests the following problem. Is any subgroup of G maximal with respect to disjointness fromp^-G pure in G 7 Or, more generally, does any such subgroup 7^ of G have the property that H r\p^ G=p^H for all ordinals (3? That is to say, is 7/^-isotype in G? We will show that indeed any such // is/?-isotype, and we will give a partial solution to the problem of determining whether any tw^o such H's are isomorphic. The foregoing considerations will lead to the solution of a more general version of the above mentioned problem of L. FUCHS.