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Problem 9
Consider a regular plane $n$-gon for $n\geq 3$. (We did this for $n=3,4$ in class.) Each way such an $n$-gon can be placed, with one covering the other, corresponds to a certain permutation of the vertices. The set of these permutations is the $n^{th}$ dihedral group $D_n$ under permutation multiplication. Find the order of the group $D_n$. Argue $\textit{geometrically}$ that this group has a subgroup having just half as many elements as the whole group has.
Solution
Since $D_n$ can be thought of as the set of all rotations and reflections of a regular plane $n$-gon, we know there are $n$ symmetries of an $n$-gon. Hence, there are $n$ reflections. Also, an $n$-gon can be rotated $n$ different ways to correspond to a certain permutation of the vertices. Hence, there are $n$ rotations. There are $2n$ rotations and reflections so $|D_n|=2n$.
Consider the rotations as a subgroup.
No matter how many times the $n$-gon is rotated, it can always be represented by a rotation (i.e. rotations composed with other rotations will yield another rotation). Hence, closed under rotations.
If we rotate an $n$-gon $n$ times where each rotation is $\frac{2\pi}{n}$ radians then we get $\rho_0$ (identity). So, each edge of the $n$-gon maps to the same edge of the $n$-gon. Hence, contains an identity.
If the $n$-gon is rotated, then there is always another rotation that can be applied to get the $n$-gon back to its initial position. Hence, each rotation has an inverse.
Therefore, the group of rotations is a subgroup of $D_n$.