# Thread: Group homomorphism

1. ## Group homomorphism

Describe a group homomorphism from $\displaystyle U_5$ to $\displaystyle S_4$.

$\displaystyle U_5$ is the group of units $\displaystyle Z/5Z$ under multiplication
$\displaystyle S_4$ is the set of permutations on 4 elements.

The hint for the exercise said to use Cayley's Theorem.

2. Originally Posted by Zennie
Describe a group homomorphism from $\displaystyle U_5$ to $\displaystyle S_4$.

$\displaystyle U_5$ is the group of units $\displaystyle Z/5Z$ under multiplication
$\displaystyle S_4$ is the set of permutations on 4 elements.

The hint for the exercise said to use Cayley's Theorem.
I think a group homomorphism is a mapping from group $\displaystyle U_5$ [with say operation $\displaystyle *$] into $\displaystyle S_4$ [with operation $\displaystyle *'$].

And $\displaystyle f(a*b)=f(a)*'f(b)$ $\displaystyle \forall a,b \in S_4$.

3. Originally Posted by Zennie
Describe a group homomorphism from $\displaystyle U_5$ to $\displaystyle S_4$.

$\displaystyle U_5$ is the group of units $\displaystyle Z/5Z$ under multiplication
$\displaystyle S_4$ is the set of permutations on 4 elements.

The hint for the exercise said to use Cayley's Theorem.
Well, what is Cayley's theorem?

The hint is telling you, basically, to work through the proof of Cayley's theorem using an example. There are a number of different homomorphisms from $\displaystyle U_5$ to $\displaystyle S_4$, but the point of this question is not to conjure up these, but to find a specific one using this theorem.

So, do you have any problems understanding the theorem? It basically says you can find a `copy' of $\displaystyle U_5$ in $\displaystyle S_4$.