# Math Help - Homomorphisms, Integers mod 6 to Symmetry gp

1. ## Homomorphisms, Integers mod 6 to Symmetry gp

There is a unique homomorphism $\theta: \mathbb{Z}_{6}\rightarrow S_{3}$ such that $\theta ([1]) =(1 \ 2 \ 3).$ Determine $\theta ([k])$ for each $[k] \in \mathbb{Z}_{6}$. Which elements are in $Ker \theta$?

I don't understand how that single condition can be used to determine the rest of the $\theta ([k])$. It doesn't seem like enough information.

2. Originally Posted by Blue Calx
There is a unique homomorphism $\theta: \mathbb{Z}_{6}\rightarrow S_{3}$ such that $\theta ([1]) =(1 \ 2 \ 3).$ Determine $\theta ([k])$ for each $[k] \in \mathbb{Z}_{6}$. Which elements are in $Ker \theta$?

I don't understand how that single condition can be used to determine the rest of the $\theta ([k])$. It doesn't seem like enough information.
If $\theta (1) = (123)$ then $\theta (2) =\theta(1+1) = (123)^2$.
And $\theta(3) = \theta(1+1+1) = (123)^3$.
And so on ...