let $G$ be dihedral group so that $D_{2n}= \langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. Prove that $N\unlhd G$ and $G/N \cong W = \{1,-1\}$

Question

let $G$ be dihedral group so that $G = D_{2n}=\langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$.

1) let $N < G$ so that $N =\langle y \rangle = \{e,y,...y^{n-1}\}$. Prove that $G \unlhd N$.

2) $G/N \cong W = \{1,-1\}$ where $W$ is a group under the normal multiplication.

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First my plan is to show that $ gng^{-1} \in N$, $\forall g,g^{-1} \in G$.

We know that $yx=xy^{n-1}$, so

$yx=xy^{n-1} \ || \ x^{-1} \cdot \iff x^{-1}yx = y^{n-1} \in N$

Any help with 2) would be great!

• what is G' (Commutator subgroup) of G

Answer 1

Hint : defifen $f: D_{2n} \to \{-1,1\}$ by, $$ f(y^i)=1, f(xy^i)=-1$$

This is homomorphism, $kerf= N$

Answer 2

Here are two different approaches for each of the questions:

Question 1

If you have a group $G$ generated by $x$ and $y$ satisfying some relations, and $H$ is a subgroup, then to show that $H$ is normal, it suffices to show that $xhx^{-1}\in H$ and $yhy^{-1}\in H$ for all $h\in H$. Prove this fact, and then do this computation for your specific group.

OR

If $H\subset G$ is a subgroup and $|H|=|G|/2$, then $H$ must be normal! To see why, use the relative sizes of $H$ and $G$ to say what a left or right coset of $H$ must look like, and deduce that every left coset is a right coset and vice-versa.

Question 2

You have two cosets, $eN$ and $xN$. Compute the multiplication table, and find an isomorphism between the quotient group and $W$.

OR

Show that EVERY group of order $2$ is isomorphic to $W$.

Answer 3

You know that $|D_{2n}|=2n$ and by definition of $N$ we get that $|N|=n$ hence by Lagrange theorem $$ |G/N|=\frac{|G|}{|N|}=2 $$

Since $N$ is also normal in $G$ we know that $G/N$ is a group.

Now, there is only one group, up to isomorphism, of order $2$ which gives the desired result.