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Group actions/operations?

  1. Nov 6, 2004 #1
    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 [tex]S_3[/tex] can act on a set [tex]X[/tex]with 4 elements.

    I mean, an action assigns with every element in [tex]S_3[/tex] 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 [tex]S_3[/tex], we assign a permutation of X. But how does that help me answer the question? :confused:
     
  2. jcsd
  3. Nov 6, 2004 #2

    matt grime

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    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).
     
  4. Nov 7, 2004 #3
    Okay, so describing all homomorphisms from [tex]S_3[/tex] to [tex]S_4[/tex] seems a little more tangible. I suppose there are four "natural" homomorphisms, described by simply ignoring one element in [tex]S_4[/tex], and using the permutation of [tex]S_3[/tex] 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.
     
  5. Nov 8, 2004 #4

    matt grime

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