Solving Homomorphism Q on Cyclic Group of Order 7

[ElDavidas]

Can anybody help me out?

"Let G be a cyclic group of order 7. Determine the number of homomorphisms from G to $S_7$"

I know the definition of a cyclic group. It's a group generated by a single element. The order 7 means that $g^7 = e$ for g within G.

I understand that a homomorphism is defined by $\Psi (ab) = \Psi(a)\Psi(b)$ for a, b that exist in G. $\Psi : G \rightarrow S_7$

My main problem is, I can't put this all together and answer the question!

Any comments/suggestions would be appreciated

 

[shmoe]

Let g be a generator of G, $$\Psi$$ a homomorphism. Since g generates the group if we know what happens to g, we know what happens to the rest of the group. So we can concentrate our efforts on g.

What can you say about the order of $$\Psi(g)$$? What then are the possibilities for $$\Psi(g)$$?

 

[ElDavidas]

What can you say about the order of $$\Psi(g)$$?

That the order of $$\Psi(g)$$ is determined by g? Purely a guess...

I know that you can calculate the order of a permutation in $S_7$ by finding the LCM of the lengths of the disjoint cycles.

Not sure if that's relevant though.

 

[shmoe]

You know $$g^7=e$$ so you know $$\Psi(g)^7=\Psi(g^7)=\Psi(e)$$

What is $$\Psi(e)$$ and what are the possibilities for the order of $$\Psi(g)$$?

 

[ElDavidas]

What is $$\Psi(e)$$ and what are the possibilities for the order of $$\Psi(g)$$?

$$\Psi(e)$$ = $$\Psi(gg^{-1})$$ = $$\Psi(g)\Psi(g^{-1})$$

I don't know how to take it further or how to come to any conclusions about this. I've been rereading my lecture notes lots of times about this area.

 

[shmoe]

$$\Psi(e)^2=\Psi(e^2)=\Psi(e)$$

You should know what $$\Psi(e)$$ must be now- not many elements in a group can equal their own square!