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?