# Thread: Homomorphism of dihedral group

1. ## Homomorphism of dihedral group

Hey I've been working on this question,

How that the following is a homomorphism
$\displaystyle \theta :{{D}_{2n}}\to {{D}_{2n}}\,\,\,givenby\,\,\,\theta ({{a}^{j}}{{b}^{k}})={{b}^{k}}\,\,\,$

$\displaystyle \theta ({{a}^{j}}{{b}^{k}})\theta ({{a}^{m}}{{b}^{n}})={{b}^{k}}{{b}^{n}}$

$\displaystyle \theta ({{a}^{j}}{{b}^{k}}{{a}^{m}}{{b}^{n}})=?$

From what it looks like it isn't a homomorphism but I'm not sure how to evaluate the last line,

It would be easy if the group commuted..

Does anyone know how to evaluate it?

I also have another question which I solved numerically but I'm not sure how to show it algebraically,

Would anyone know how to show
$\displaystyle \text{Remaider}\left( \frac{ab}{n} \right)\ne \text{Remainder}\left( \frac{a}{n} \right)\times \text{Remainder}\left( \frac{b}{n} \right) For\,\,a,b,n\in \mathbb{Z}$

Or would it be sufficient to just know that its not true?

2. ## Re: Homomorphism of dihedral group

The dihedral group $\displaystyle D_{2n}$ consists of $\displaystyle n$ rotations and $\displaystyle n$ reflections; the rotations form a cyclic subgroup $\displaystyle <a>$ of order $\displaystyle n.$ An element of $\displaystyle D_{2n}$ can be written as $\displaystyle a^jb^k$ where $\displaystyle j\in\mathbb Z,$ $\displaystyle b$ is any reflection and $\displaystyle k=0$ or $\displaystyle 1.$

A rotation generally does not commute with a reflection; however, any product of a reflection followed by a rotation is equal to a product of a rotation followed by another reflection. In other words, $\displaystyle ba^m=a^{m'}b$ for some other $\displaystyle m'\in\mathbb Z.$ Hope this helps.

For your second question, if you take $\displaystyle a=b=6$ and $\displaystyle n=4,$ then $\displaystyle 0\ne2\times2.$

3. ## Re: Homomorphism of dihedral group

Thanks Sylvia it definitely helped

4. ## Re: Homomorphism of dihedral group

Sorry to bug you again,

I thought I had the question down but then I realised I made an assumption that the group binary operation was multiplication,

The question was show the following is a homomorphism
$\displaystyle \varphi :\mathbb{Z}\to {{\mathbb{Z}}_{n}}\,\,\,\,\,where\,\,\varphi (a)=\text{Remainder}\left( \frac{a}{n} \right)$

When I did this step I assumed multiplication,

$\displaystyle \varphi (a)\varphi (b)=\text{Remainder}\left( \frac{a}{n} \right)\times \text{Remainder}\left( \frac{b}{n} \right)$

but I now I'm paranoid that it should be addition,

$\displaystyle \varphi (a)+\varphi (b)=\text{Remainder}\left( \frac{a}{n} \right)+\text{Remainder}\left( \frac{b}{n} \right)$

Since in class we mainly use the integers under an addition modulo rather then multiplication, we only do multiplication on integers co-prime to the dimension

If n = 4 in this case then would the operation be addition modulo ten then 0 = 2+2=4 with a=b=6 like you said before?

I read online somewhere that "Reduction modulo n is a homomorphism" is that true?

I found some cases by mucking around in mathematica where its not satisfied under addition, if n=5 a=11 b=8,

I should of posted the question up in the first place haha

Sorry again for pestering you once more

5. ## Re: Homomorphism of dihedral group

Originally Posted by Jesssa
The question was show the following is a homomorphism
$\displaystyle \varphi :\mathbb{Z}\to {{\mathbb{Z}}_{n}}\,\,\,\,\,where\,\,\varphi (a)=\text{Remainder}\left( \frac{a}{n} \right)$
This follows from the fact that if $\displaystyle a\equiv r_1\mod n$ and $\displaystyle b\equiv r_2\mod n$ then $\displaystyle a+b\equiv r_1+r_2\mod n.$

6. ## Re: Homomorphism of dihedral group

phi(a+b) = Rem[(a+b)/n] = Rem[a/n]+Rem[b/n] = phi(a) + phi(b) !

Thanks again Sylvia!

7. ## Re: Homomorphism of dihedral group

here is a (somewhat more) formal proof. suppose a = a' (mod n), and b = b' (mod n). then a+b = a'+b' (mod n).

since a = a' (mod n), by definition a - a' = kn, for some integer k. we can re-write this as a = a' + kn.

similarly, since b = b' (mod n), then b - b' = mn, for some integer m. we can re-write this as b = b' + mn.

therefore: a + b = (a' + kn) + (b' + mn) = a' + b' + (k + m)n. thus (a + b) - (a' + b') = (k + m)n, so

a + b = a' + b' (mod n) (since k+m is an integer whenever k and m are).

in particular, if [a] is the unique element of Z congruent to a mod n between 0 and n-1, then [a] + [b] = [a + b],

which makes the map a-->[a] a homomorphism from Z to Zn: φ(a + b) = [a + b] = [a] + [b] = φ(a) + φ(b).

8. ## Re: Homomorphism of dihedral group

In fact

Originally Posted by Jesssa
When I did this step I assumed multiplication,

$\displaystyle \varphi (a)\varphi (b)=\text{Remainder}\left( \frac{a}{n} \right)\times \text{Remainder}\left( \frac{b}{n} \right)$
it is also true that $\displaystyle \varphi(ab)=\varphi(a)\varphi(b)$ since it is also true that $\displaystyle a\equiv a'\mod n$ and $\displaystyle b\equiv b'\mod n$ $\displaystyle \implies$ $\displaystyle ab\equiv a'b'\mod n.$ (NB: While $\displaystyle 0\ne2\times2,$ that's only in the ring of integers; it will be true if you reduce modulo $\displaystyle 4.)$ Thus $\displaystyle \varphi$ is a ring homomorphism, not just a homomorphism between additive groups.