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Let $(G, *)$ be a group, let $e_G$ be the identity in $G$, and let $H$ be a non-empty subset of $G$. Can it happen that $(H, *)$ be a group in its own right (with the same binary operation) but with an identity element $e_H$ that is different from $e_G$? Of course, our requirement is that $e_G$ is not in $H$, for otherwise $e_G$ will have to be the same as $e_H$.

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No: For if we choose any $h \in H$, we see that

$$e_H h = h$$

Now by cancellation in $G$, we find that $e_H = e_G$.

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