this Question was asked before on this site:
Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order 2 in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $n = |xy|.$
A user pointed out that we can replace $xy$ by the rotation element and $y$ by the reflection element and do a homomorphism between the groups and show that they are isomorphic. What does the element $x$ map to?
I know the reflection element has order $2$ but what other element in the dihedral group has order $2$?
Another Related question on site: Dihedral group – elements of order $2$
If our polygon has odd number of edges, shouldn’t our polygon have only one element of order $2$? (Just the reflection?)
Best Answer
The keep this question from being unanswered...
Let $G$ be a finite group generated by $x$ and $y$, with $x$ and $y$ of order $2$. We want to show that $G\cong D_{2n}$ (the dihedral group of order $2n$), where $n$ is the order of $xy$.
To answer your questions first: in the dihedral group $D_{2n}=\langle r,s\mid r^n = s^2 = 1, sr=r^{-1}s\rangle$, every element not in $\langle r\rangle$ is of order $2$. To verify this, note that every element can be written uniquely as $r^is^j$, with $0\leq i\lt n$, $0\leq j\lt 2$. The elements not in $\langle r\rangle$ are precisely the ones with $j=1$. Such an element satisfies: $$\begin{align*} (r^is)^2 &= r^i(sr^i)s\\ &= r^i(r^{-i}s)s &\text{(since }sr=r^{-1}s\text{)}\\ &= r^0s^2\\ &= 1. \end{align*}$$ Thus, all such elements are elements of order $2$.
When $n$ is odd, these are the only elements of order $2$; when $n$ is even, all of these are elements of order $2$, and so is $r^{n/2}$. So in a dihedral group, you always have at least half the elements of order $2$.
If you think of the dihedral group as the symmetries/rigid motions of a regular $n$-gon sitting on the plane inscribed in the unit circle, you have several axes through which you can reflect the polygon, not just the $x$-axis. The bisection through each vertex gives you a line through which you can reflect the polygon, getting an element of order $2$.
Now, the proof of the desired statement. We note that $xy$ and $y$ satisfy the relations in the presentation of $D_{2n}$: indeed, by definition of $n$ we know that $(xy)^n = 1$; and $y^2=1$ by assumption. Finally, we have that $$\begin{align*} y(xy) &= (yx)y\\ &= (y^{-1}x^{-1})y &\text{(since }x^2=y^2=1\text{)}\\ &= (xy)^{-1}y. \end{align*}$$ By von Dyck's Theorem there is a homomorphism $f\colon D_{2n}\to G$ mapping $r$ to $xy$ and $s$ to $y$. Under this homomorphism, $x$ is the image of $rs$.
(There are other possible homomorphisms, since the map sending $r\mapsto r^i$, $s\mapsto s$, with $\gcd(i,n)=1$, is an automorphism of $D_{2n}$, so pre-composing it with $f$ gives you a slightly different map).
The link you give in your question contains several different answers showing that this $f$ is indeed an isomorphism.