Group Actions on Sets: Understanding the Permutation Group S_3

[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 Xwith 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?