Consider a regular 2$n$-gon $P_{2n}$ with vertices denoted $a_1, a_2,...,a_n$, by going in the clockwise orientation. Consider the subset of vertices of $P_{2n}$ given by $S$ = {$a_2,a_4,a_6,...a_{2n-2},a_{2n}$}. A symmetry $\gamma$ in $D_{2n}$ is called $S-admissible$ if for all $x \in S$, we have $\gamma(x) \in S$. Prove that
$H =$ {$\gamma \in D_{2n} \mid \gamma$ is $S$-admissible} is a subgroup of $D_{2n}$. Explain in words why $H$ has to be isomorphic to $D$
So H is a subgroup where the rotations in $H$ are only to even-numbered vertices. Since $S$ has rotations to all vertices, $H$ is obviously a subgroup of $S$. I'm not sure if I am 100% right on this because it feels like it is too simple for the question.
For the isomorphism part, isn't it an isomorphism because $H$ is a subgroup of $S$?