Consider the symmetric group $ \langle S_{10}, ◦ \rangle$
Show that $H = \{ \sigma \in S_{10} : \sigma(5) = 5\}$ is a subgroup of $S_{10}$
By the subgroup test a subset H of G is a subgroup if and only if it satisfies the following:
- The identity $e$ of $G$ is in $H$
- If $h_1, h_2 \in H$ then $h_1h_2 \in H$
- If $h \in H$ then $h^{-1} \in H$
But I have no idea how to apply that to this example, not for a lack of trying. I have a test on group theory soon and really need help understanding this problem. Could someone please explain how to show one of the conditions and then at least I can have a fresh attempt at the others. Thanks.