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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- Oct 21st 2010, 06:05 PMkathrynmathShowing 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).

- Oct 22nd 2010, 12:11 AMSwlabr
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 $\displaystyle \sigma_1 \sigma_2$.