# Showing something is a subgroup

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• Oct 21st 2010, 06:05 PM
kathrynmath
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).
• Oct 22nd 2010, 12:11 AM
Swlabr
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 $\sigma_1 \sigma_2$.