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accepted Is there a problem with this example?
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comment Let $A,B$ be two subgroups of $G$. Show that $g(A\cap B)=gA \cap gB$ for all $g \in G$
This would be easier to think about if you took $c \in A \cap B$. So this would rephrase your first step to $gc \in g(A \cap B)$. The result basically follows from here.
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