Given the dihedral group $D_n$, where $r$ is a single rotation, and $s$ is a reflection, I must show that $s \circ r \circ s = r^{-1}$.
$D_n$ is a group, with properties:
1) $1 = r^0 = s^0$
2) for any $k \in \mathbb{Z}$, $s^k = s^{k \mod 2}$, for any $i, j \in \mathbb{Z}$, $s^{i} \circ s^{j} = s^{i + j} = s^{i + j \mod n}$
3) for any $k \in \mathbb{Z}$, $r^k = r^{k \mod n}$, for any $i, j \in \mathbb{Z}$, $r^{i} \circ r^{j} = r^{i + j} = r^{i + j \mod n}$
With these properties alone, I have not been able to show that $s \circ r \circ s = r^{n - 1} = r^{-1}$. Thus, either I have simply not hit upon the right ideas, or additional properties are required. I suspect additional properties are required because I do not have any that talk about $s^i \circ r^j$ and $r^j \circ s^i$.
But what additional properties should I have for the Dihedral group? One idea I had:
4) for any $i, j \in \mathbb{Z}$, $s^i \circ r^j = r^t$ and $r^j \circ s^i = r^u$ for some $0 \leq t, u < n$ (since $r^k = r^{k \mod n}$ by property 3)
But this property seems too weak to give me what I am looking for. Should it be strengthened with something more specific, e.g.:
4') for any $i, j \in \mathbb{Z}$, $s^i \circ r^j = s^{i - j}$ and $r^j \circ s^i = s^{i + j}$
What is the minimal amount of information needed to define a dihedral group purely algebraically?