# Math Help - prove homomorphism and determine Kernel

1. ## prove homomorphism and determine Kernel

Phi is the mapping from Z to Z_12 by for all a that is a member of Z, phi(a)=[3a]mod 12.
-Prove that phi is homomorphism
-Determine Ker(phi)

This is what I have, but don't think it is right, can someone help me Please?

Homomorphism:
phi(ab)=(ab)^12=a^12 * b^12=phi(a)phi(b)

And the Ker(phi)=<cos 120 + i sin120>

I am completely lost and dont know if I did it right!?

2. Originally Posted by mandy123
Homomorphism:
phi(ab)=(ab)^12=a^12 * b^12=phi(a)phi(b)
You have $\phi (ab) = [3ab]_{12}$ and $\phi(a) \phi(b) = [3a]_{12} \cdot [3b]_{12}$.
To be a homomorphism you need $[3a]_{12}\cdot [3b]_{12} = [3ab]_{12}$.
This seems to be work, therefore it cannot be a homomorphism.

3. What about under addition would it be homomorphism then?

What about the Ker(phi) under the operation addition?

4. Originally Posted by mandy123
What about the Ker(phi) under the operation addition?
By definition $\ker (\phi) = \{ x\in \mathbb{Z} : [3x]_{12} = [0]_{12} \}$.

Therefore, $3x\equiv 0 (\bmod 12) \implies x\equiv 0 (\bmod 4)$.

5. Thank you so much, I am starting to understand this stuff!!!!