Solution
(related to Problem: Verifying Subgroup Properties)
There are two possibilities to show that a nonempty subset $H$ of a group $(G,\ast)$ is its subgroup. Both possibilities are equivalent. Which possibility you choose is more or less a matter of taste and of simplicity.
$(1)$ Verify the subgroup properties
 Demonstrate that $e\in H,$ where $e$ is also the neutral element of $G$ with respect to the operation $"\ast".$
 Demonstrate that if $a\in H,$ then also its inverse $a^{1}\in H.$
 Demonstrate that $H$ is closed under the operation $"\ast",$ i.e. $a\ast b\in H$ for all $a,b\in H.$
I.e. show that $a\ast b^{1}\in H$ for all $a,b\in H.$
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