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Math Help - Showing something is a subgroup

  1. #1
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    Showing something is a subgroup

    Let S be a set and let a be a fixed element of S. Show that s is an element of Sym(S) such that s(a)=a is a subgroup of Sym(S).
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  2. #2
    MHF Contributor Swlabr's Avatar
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    If \sigma_1, \sigma_2 are permutations which fix an element a \in X, \sigma_i \in S_X, then you need to prove two things,

    i. \sigma_1^{-1} also fixes a.

    ii. \sigma_1 \sigma_2 also fixes a.

    Can you work out why these two things are sufficient?

    Now, to prove i. you should note that \sigma_1: a \mapsto a. If \sigma^{-1}: a \mapsto b then where must b be sent to in (\sigma_1^{-1})^{-1} = \sigma_1?

    To prove ii. simply plug in a into \sigma_1 \sigma_2.
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