Let $\displaystyle A$ be an abelian group.

We know that all subgroups of $\displaystyle A$ are normal in $\displaystyle A$.

If $\displaystyle A$ is a free abelian group, are cyclic subgroups of $\displaystyle A$ also normal in $\displaystyle A$?

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- Jul 25th 2009, 10:32 PMdeniselim17cyclic subgroup in free abelian group
Let $\displaystyle A$ be an abelian group.

We know that all subgroups of $\displaystyle A$ are normal in $\displaystyle A$.

If $\displaystyle A$ is a free abelian group, are cyclic subgroups of $\displaystyle A$ also normal in $\displaystyle A$? - Jul 25th 2009, 10:43 PMdeniselim17
I think the answer is yes.

Since free abelian group is also an abelian group, then $\displaystyle bh=hb$ for all $\displaystyle b \in B, h\in <h>$ because $\displaystyle <h> \leq B$.

So we can get $\displaystyle b^{-1}hb=h \in <h>$.