Belian group A that is the direct sum of cyclic groups

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SUMMARY

The discussion centers on the annihilator of an abelian group A, specifically when A is expressed as a direct sum of cyclic groups, such as A = C5 ⊕ C35. It is established that the annihilator of A, viewed as a Z-module, is indeed generated by the elements (5, 35). The importance of expressing the group in a form where each subscript divides the following is emphasized, as it simplifies the process of determining the annihilator.

PREREQUISITES
  • Understanding of abelian groups and their properties
  • Knowledge of cyclic groups and their structure
  • Familiarity with Z-modules and annihilators
  • Basic concepts of group theory and direct sums
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If I have an abelian group A that is the direct sum of cyclic groups, say
A=[tex]C_5 \oplus C_35[\tex], would I be right in saying the annihilator of A (viewed as a Z-module) is generated by (5,35)? If not, how do I find it?[/tex]
 
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ought it not to be an element of Z, which we presume is acting diagonally, and which is thus 35?
 
I understand now, thanks. Now I also understand why you'd put the group into a form where each subscript divides the following, instead of just leaving it in primary form. It makes it much easier to find the annihilator.
 

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