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We had the statement that with an exact sequence of multiplicatively written abelian groups $U \mapsto V \mapsto W \mapsto X \mapsto Y$ and in $U$, $V$, $X$, $Y$ every group element has a unique $n$-th root ($n \in \mathbb{Z}$) That $W$ has this property as well.

How do I see that? Does it help me that the $n$ roots form a group under multiplication? Or why is that? Any help/hint/tipp would be great :)

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1 Answer 1

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Raising to the $n$th power is a homomorphism of an abelian group to itself. Apply the five-lemma to the commutative diagram consisting of two rows of your given exact sequence, where all of the vertical maps are raising to the $n$th power.

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