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Thread: 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 $\displaystyle \sigma_1$, $\displaystyle \sigma_2$ are permutations which fix an element $\displaystyle a \in X$, $\displaystyle \sigma_i \in S_X$, then you need to prove two things,

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

    ii. $\displaystyle \sigma_1 \sigma_2$ also fixes $\displaystyle a$.

    Can you work out why these two things are sufficient?

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

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