# Thread: Homomorphisms, Integers mod 6 to Symmetry gp

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

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

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

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

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