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).
If , are permutations which fix an element , , then you need to prove two things,
i. also fixes .
ii. also fixes .
Can you work out why these two things are sufficient?
Now, to prove i. you should note that . If then where must be sent to in ?
To prove ii. simply plug in into .