Does This Mapping Define a Group Homomorphism from ℤ_n to D_n?

[spacetimedude]

Homework Statement

$ℤ_n → D_n$ sending z modn → $g^z$ where g is rotation by an nth of a turn.

Homework Equations

Group homomorphism imply $θ(g_1*g_2)=θ(g_1)*θ(g_2)$

The Attempt at a Solution

Before anything, I'd like to know if Group homomorphism imply $θ(g_1+g_2)=θ(g_1)$x$θ(g_2)$ I've seen $θ(g_1$x$g_2)=θ(g_1)×θ(g_2)$ and $θ(g_1+g_2)=θ(g_1)+θ(g_2)$ but not $θ(g_1+g_2)=θ(g_1)$x$θ(g_2)$. Can the operator * be different on the left and right side?

Attempt:
$θ(z_1+z_2)$=g^{$z_1+z_2 modn$}
I'm not sure how to go from here. I'm sure I need to use the fact that $g^n=e$ but I don't know how to proceed.
Any help will be appreciated!

 

[LCKurtz]

Homework Statement

$ℤ_n → D_n$ sending z modn → $g^z$ where g is rotation by an nth of a turn.

What is $D_n$? And what is a "turn" and an nth of a turn?

Homework Equations

Group homomorphism imply $θ(g_1*g_2)=θ(g_1)*θ(g_2)$

The Attempt at a Solution

Before anything, I'd like to know if Group homomorphism imply $θ(g_1+g_2)=θ(g_1)$x$θ(g_2)$ I've seen $θ(g_1$x$g_2)=θ(g_1)×θ(g_2)$ and $θ(g_1+g_2)=θ(g_1)+θ(g_2)$ but not $θ(g_1+g_2)=θ(g_1)$x$θ(g_2)$. Can the operator * be different on the left and right side?

Yes, they are usually different groups. Look up the definition of group homomorphism.

 

[Office_Shredder]

If you are having trouble with the intuition, you should try picking a value of n, say n=3 to start, and just explicitly write out every possible pair of elements in $\mathbb{Z}_n$ and how they add up and what they would get mapped to in $D_n$.

 

[vela]

The Attempt at a Solution

Before anything, I'd like to know if Group homomorphism imply $θ(g_1+g_2)=θ(g_1)$x$θ(g_2)$ I've seen $θ(g_1$x$g_2)=θ(g_1)×θ(g_2)$ and $θ(g_1+g_2)=θ(g_1)+θ(g_2)$ but not $θ(g_1+g_2)=θ(g_1)$x$θ(g_2)$. Can the operator * be different on the left and right side?

Yes, the operations generally are different, even if you happen to use the same symbol for them. When you wrote $\theta(g_1+g_2) = \theta(g_1)+\theta(g_2)$, the + on the lefthand side isn't the same as the + on the righthand side because $g$ and $\theta(g)$ are from two different sets.