# Group actions/operations?

## Main Question or Discussion Point

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? Related Linear and Abstract Algebra News on Phys.org
matt grime
Homework Helper
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).

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