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Math Help - Kernel of a group action and Sa

  1. #1
    Super Member Bernhard's Avatar
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    Kernel of a group action and Sa

    Dummit and Foote Section 1.7 Exercise 5 reads as follows:

    Prove that the kernel of an action of the group G on a set A is the same as the kernel of the corresponding permutation representation G \longrightarrow S_A

    I am having real trouble with this - can anyone help me?

    By the way - what is the meaning of "the corresponding permutation representation \longrightarrow S_A" ???

    I would be really grateful for some assistance.

    Peter
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  2. #2
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    Re: Kernel of a group action and Sa

    there are two ways to look at a group action.

    1)as a "multiplication" GxA-->A given by (g,a)-->g.a (also written as g(a)).

    2) as a homomorphism φ from G--->Sym(A) (the set of bijections on A, so each element of g maps to a permutation \sigma_g of A).

    now, the kernel as per the first way of looking at a group action is the set {g in G: g.a = a, for all a in A}.

    the kernel in the second way of looking at a group action is the set {g in G : \sigma_g = id_A, the identity function on A}

    we want to show these are the same thing. so suppose g.a = a, for all a in A. then
    \sigma_g(a) = a for all a in A, that is \phi(g) = \sigma_g = id_A, so g \in ker(\phi).

    on the other hand, suppose g \in ker(\phi). then g.a = \sigma_g(a) = \phi(g)(a) = id_A(a) = a, so that g is in the set {g in G: g.a = a, for all a in A}.

    the whole point of this exercise is to show you the two definitions of a group action "really are the same".
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