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Finding permutations of a stabilizer subgroup of An

  1. Oct 15, 2012 #1
    Alright, I understand what a stabilizer is in a group, and I know how to find the permutations of An for any small integer n, but for a stabilizer, since it just maps every element to 1, would all permutations just be (1 2) (1 3) ... (1 n) for An?
     
  2. jcsd
  3. Oct 25, 2012 #2
    I think you may be confused about what the stabilizer is.

    Suppose you have
    1. a group G
    2. a set S (this set is not necessarily a group, it is just a bunch of elements)

    There need be no connection between the elements of S and the elements of G. That is, S is just a collection of elements that can be entirely distinct from the elements of G.

    An action of G on S is a map from [itex]G \times S \rightarrow S[/itex]. (We also place two requirements on the behavior of this map, but just ignore this for the moment). In other words, "G acts on S" means that given any [itex]g \in G[/itex] and any [itex] x \in S[/itex] we can "apply" g to x and get a new element, y, of S. Symbolically, gx = y (though this looks like we are operating g and x using the group operation of G, this is not what we are doing. x is not even in G; it is in S).

    The group of Sn of permutations provides a very natural example of all this. Take for instance S3. Here G = S3 and S = {1,2,3}. Given a particular permutation [itex]\sigma \in S_3[/itex], we can talk about what the permutation does to any element of S. Take [itex]\sigma = [/itex] 'the permutation that transposes 1 and 2'. Then [itex]\sigma2 = 1[/itex]. So [itex]\sigma[/itex] "acts" on the element 2 and gives the element 1.

    Are there any permutations in [itex]S_3[/itex] which act on 2 and just give 2? Yes, there are two of them:
    1. [itex]\sigma_1[/itex] = 'the permutation that transposes 1 and 3'
    2. [itex]\sigma_2[/itex]= 'the permuation that leaves all the elements fixed' (identity)

    The stablilzer of 2 is the set of both these permutations: [itex] Stab(2) = \{\sigma_1, \sigma_2\} [/itex]. In general, the stablilizer of an element [itex] x \in S [/itex] is:
    [tex]Stab(x) = \{g \in G | gx = x\}.[/tex]

    Here is where I think you are somewhat confused. The stablilzer does not map every element to 1. The stablilzer of [itex]x \in S[/itex] consists of all the elements of G that send x to x. The stablilzer does not send anything to 1 because there isn't really a 1 in S. 1 is in the group G.
     
    Last edited: Oct 25, 2012
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