CHAPTER XV - DECOMPOSITIONS INTO DIRECT SUMS OF SUBSETS

Abstract

This chapter discusses decompositions into the direct sums of subsets. The direct sum of subgroups, also the direct sum of subsets, may be extended to the case when the number of components is not finite. The chapter discusses direct decompositions with infinitely many cyclic subsets as components. A subset P of a group G is periodic if and only if it is of the form P ={g} + K for some nonzero element g and for some subset K of G. If Q is a weakly periodic, but not periodic subset of G and is at the same time a quasi-direct summand of G, then it is of the form Q = a + C. Here, C = [g]m is a cyclic subset of G such that m divides the order of g in case g is of finite order.