Hw5 Problem 11

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Problem 11

(*) Suppose that $H$ is a nonempty subset of a group $G$ that is closed under the group operation and has the property that if $a$ is not in $H$ then $a^{-1}$ is not in $H$. Is $H$ a subgroup?

Solution

Since $H$ is nonempty there must be some $a \in H$. Now we know that $a^{-1} \in H$, otherwise $(a^{-1})^{-1} = a \notin H$ by the given property. So then $a^{-1} \in H$ for all $a \in H$. Now since $H$ is closed under the group operation $aa^{-1} = e \in H$. Then $H$ is closed under the group operation, contains $e$, and contains $a^{-1}$ for all $a \in H$. Therefore $H$ is a subgroup of $G$.