Question
Find a subgroup of $Z_{4} \oplus Z_{2}$ that is not of the form $H \oplus K$, where $H$ is a subgroup of $Z_{4}$ and $K$ is a subgroup of $Z_{2}$.
Step 1
They are: $$(0,0), (1,0), (2,0), (3,0), (0,1), (1,1), (2,1), (3,1)$$ Now, let's consider the subgroup $H = \{(0,0), (2,0), (0,1), (2,1)\}$. We can verify that this is indeed a subgroup of $Z_4 \oplus Z_2$: Show more…
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Let $G$ be an abelian group. Let $H$ be a subgroup of $G$, and let $K=\left\{x \in G: x^{2} \in H\right\}$. Prove that $K$ is a subgroup of $G$.
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