1. ## Cyclic Group

Hello i was wondering if anybody could help me with the following question:

Let (G,·) be a group. Given a group action s : G → S4 having the property thats(f) = (1 2) and s(g) = (3 4) for certain elements f,g ∈ G. Can (G,·) be a cyclicgroup?
(Recall that a group (G,·) is called cyclic if there exists γ ∈ G such thathγi, the subgroup generated by γ, is equal to G.)

As I am completely lost.

I think that the argument is something like there doesn't exist any number $\displaystyle a$ such that $\displaystyle (12)^a = (34)$, but I am not sure.

King Regards

2. ## Re: Cyclic Group

As is standard, I assume you mean that s is a homomorphism from G into $S_4$. Can G be cyclic? No. If G were cyclic, then G/Ker(s) would be cyclic and would then contain two different elements of order 2 -- it is "easy" to see this is impossible in a cyclic group.

3. ## Re: Cyclic Group

Thank you for the reply. I am still having problem realizing when to use results regarding the quotient group.