A STUDY ON DIRECT SUMS OF ABELIAN GROUPS

Author (s) : V.VIMALA , Mrs.V.JOTHILAKSHMI, Dr.KANTHAKUMAR

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Abstract :
If G is an abelian group, and _𝐽is an indexed family of   subgroups of G. We say that the groups 𝐺_𝛼 generate G if every element x of G can  be written as a finite sum of elements of the groups 𝐺_𝛼 Since G is abelian, we can  always rearrange such a sum to group together terms that belong to a single 𝐺_𝛼;  hence we can always write x in the form   X = 𝑋_𝛼1 +……….. +𝑋_𝛼𝑛 where the indices 𝛼_𝑖; are distinct. In this case, we often write X as the formal sum X = _𝐽 𝑋_𝛼,where it is understood that 𝑋_𝛼 = 0 if α is not one of the indices 𝛼₁…….𝛼_𝑛. If the groups 𝐺_𝛼 generate G, we often say that G is the sum of the groups 𝐺_𝛼. writing G =  _𝑗 𝑋_𝛼 in general, or G = 𝐺₁ + ... + 𝐺_𝑛 in the case of the finite index set {1, ...,  n}.Now suppose that the groups Ga generate G, and that for each x  G, the expression  X =  𝑥_𝛼 for x is unique. That is, suppose that for each X  G, there is only one J tuple (_𝐽 with 𝑋_𝛼 = 0 for all but finitely many a such that X = . Then G is said to be the direct sum of the groups  , and we can written as G =  αJ or in the finite case, G = 𝐺₁⨁…. . A space X is paracompact if every open covering A of X has a locally finite  open refinement B that covers X. .Many authors, following the lead of Bourbaki, include as part of the definition  of the term para compact the requirement that the space be Hausdorff. We shall not  follow this convention.

Keywords : Compact, Paracompact, Abelian Group,Covering

Conference Date : 13/11/2020

Pages : 100

Paper ID : 20100