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Is this true that a pure subgroup of divisible group is also divisible?

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The statement of your question itself would make a better title, even if it looks strange to have the statement just repeat the title. –  Stefan Walter Jun 6 '11 at 9:53
    
@Stefan: Thanks for the advice. I made it a bit better. –  Sama Jun 6 '11 at 10:01

1 Answer 1

This follows directly from the definitions. A group $G$ is divisible if ever element has an $n$th root for every $n$. A subgroup is $H$ pure if, whenever an element of $H$ has an $n$th root in $G$, it has one in $H$. So let $H$ be a pure subgroup of a divisible group $G$. Given any $h\in H$ and $n\in \mathbb{N}$, $h$ has an $n$th root in $G$, and hence in $H$. Since this holds for every $h$ and $n$, every element of $H$ is divisible in $H$.

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