Let [tex]n[/tex] be a nonzero integer. An abelian group [tex]A[/tex] is called [tex]n[/tex]-divisible if for every [tex]x \in A[/tex], there exists [tex]y \in A[/tex] such that [tex]x=ny[/tex]. An abelian group [tex]A[/tex] is called [tex]n[/tex]-torsionfree if [tex]nx=0[/tex] for some [tex]x \in A[/tex] implies [tex] x=0[/tex]. An abelian group [tex]A[/tex] is called uniquely [tex]n[/tex]-divisible if for any [tex]x \in A[/tex], there exists exactly one [tex]y \in A[/tex] such that [tex]x=ny[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]\mu_n : A \rightarrow A[/tex] be the map [tex]\mu_n(a)=na[/tex]

(a) Prove that [tex]A[/tex] is [tex]n[/tex]-torsionfree iff [tex]\mu_n[/tex] is injective and that [tex]A[/tex] is [tex]n[/tex]-divisible iff [tex]\mu_n[/tex] is surjective.

(b) Now suppose [tex]0 \rightarrow A \overset{f}{\rightarrow} B \overset{g}{\rightarrow} C \rightarrow 0 [/tex] is an exact sequence of abelian groups. It is easy to check that the following diagram commutes:[see attachment]

Suppose that [tex]B[/tex] is uniquely [tex]n[/tex]-divisible. Prove that [tex]C[/tex] is [tex]n[/tex]-torsionfree if and only if [tex]A[/tex] is [tex]n[/tex]-divisible.

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# N-divisible, n-torsionfree

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