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Problem 53 (Finite Subgroup Test)

Let $G$ be a group. Suppose that $H$ is a nonempty finite subset of $G$ and that $H$ is closed under the operation of $G$ (so if $a,b\in H$, then we must have $ab\in H$). Prove that $H$ is a subgroup of $G$.

Hint. If you use the subgroup test (show that $H$ is a nonempty subset of $G$ that is closed under the operation and taking inverses), then we get to assume it's a nonempty subset of $G$ that is closed under the operation. All you have to do is explain why it's closed under taking inverses. The work you did in the problem The Inverse In A Finite Group Is A Power Of The Element should help you quite a bit. However, you can't use this theorem directly because you do not know that $G$ is a finite group. You'll want to reuse your work from that problem, not refer to the problem.

Solution

Let $G$ be a group. Suppose $H$ is a closed nonempty finite subset of $G$.

We will use the subgroup test to show $H$ is a subgroup of $G$. By construction, we know $H$ is nonempty. Similarly, we know $H$ is a subset of $G$. Also by construction, we know $H$ is closed under the operation of $G$. We must show $H$ is closed under taking inverses.

Pick $a\in H$.

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