Question
Suppose that for each prime $p, Z_{p}$ is the homomorphic image of a group $G .$ What can we say about $|G| ?$ Give an example of such a group.
Step 1
Since $Z_p$ is the homomorphic image of $G$ for each prime $p$, there exists a homomorphism $f_p: G \to Z_p$ for each prime $p$. Show more…
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Let $G$ be an abelian group. Let $H$ be a subgroup of $G$, and let $K$ consist of all the elements $x$ in $G$ such that some power of $x$ is in $H$. That is, $K=\left\{x \in G:\right.$ for some integer $\left.n>0, x^{n} \in H\right\}$. Prove that $K$ is a subgroup of $G$.
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