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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- October 21st 2010, 07:05 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).

- October 22nd 2010, 01:11 AMSwlabr
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 .