How do S_3 group actions on a 4-element set work?

[T-O7]

Okay, so I'm trying to understand the notion of group actions, and I'm having a little difficulty understanding how to work on this question:

Describe all the ways the group $$S_3$$ can act on a set $$X$$with 4 elements.

I mean, an action assigns with every element in $$S_3$$ a permutation of the set X. The confusing thing for me now is that the group we start with is a permutation group itself, so it's like for every permutation in $$S_3$$, we assign a permutation of X. But how does that help me answer the question?

 

[matt grime]

You are asked to describe all homomorphisms from S_3 to S_4.

One way to do this is to pick generators of S_3 in a suitable fashion.

We may use the fact it is generated by transpositions.

S_3 is generated by (12) and (23)

How cany you embed these elements in S_4 in a group homomorphic way?

Hint: if phi is a homomorphism, ord(phi(x)) divides ord(x).

 

[T-O7]

Okay, so describing all homomorphisms from $$S_3$$ to $$S_4$$ seems a little more tangible. I suppose there are four "natural" homomorphisms, described by simply ignoring one element in $$S_4$$, and using the permutation of $$S_3$$ to permute the remaining 3 elements. Hmm...by the looks of it, I don't think there can be any other homomorphism, but I'm now thinking of a way to show that.

 

[matt grime]

Well, there are other homomophisms; who said the map needed to be injective?