CHAPTER IV - DIRECT SUMMANDS AND PURE SUBGROUPS

Abstract

Hitherto we have got acquainted with groups which decompose into the direct sum of cyclic or quasicyclic and full rational groups, i. e. of groups whose structure is very simple and is completely known. Most groups cannot, however, be so decomposed, but many of them may be written as a direct sum of two of their subgroups one of which has more or less special properties. In this chapter we study therefore the direct summands and then turn our attention to the consideration of pure subgroups which constitute a generalization of the concept of direct summands. Pure subgroups will continually occur in dealing with abelian groups, for it is not hard to verify, in a definite case, the existence of pure subgroups subject to certain requirements, and there exists a lot of criteria assuring the direct summand property of pure subgroups. Besides this methodical importance, pure subgroups have also the advantage of being a proper substitute for direct summands in a number of cases when direct summands cannot be obtained.