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Proof Jumble: Subgroup Intersection, \(c\in H\cap K\) and \(d\in H\cap K\)…
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Let \( (H, \star)\) and \((K,\star)\) be subgroups of some group \((G,\star)\).
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First, we will show that \(H\cap K\) is a subset of \(G\).
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Since \(x\) is an element of \(H\cap K\), \(x\) is an element of \(H\).
Since \(H\) is a subgroup of \(G\), \(H\) is a subset of \(G\) forcing \(x\) to be an element of \(G\).
Therefore every element of \(H\cap K\) is also an element of \(G\). Hence \(H \cap K\) is a subset of \(G\).
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For \( (H\cap K,\star)\) to also be a subgroup of \((G,\star)\), \(H\cap K\) must be a subset of \(G\), \( H\cap K\) must be closed under \(\star\), and \(H\cap K\) must be closed under inverses.
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By definition, every element of \(H\cap K\) is an element of both \(H\) and \(K\).
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Next, we will show that \(H\cap K\) is closed under \(\star\).
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Finally, we will show \(H\cap K\) is closed under inverses.
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Therefore, \((H \cap K, \star)\) is a subgroup of \((G,\star) \) , and any intersection of two subgroups of a group is itself a subgroup of that group. \(\blacksquare\)
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