1. ## prove for subgroup

Let X be a set, and let Y⊆X. Show that the subset of S_{x} consisting of all f such that f(y)=y for all y∈Y forms a subgroup

2. Originally Posted by hzhang610
Let X be a set, and let Y⊆X. Show that the subset of S_{x} consisting of all f such that f(y)=y for all y∈Y forms a subgroup
You need to confirm all the properties for being a subgroup are satisfies. Show what you did.

3. ## I did the first part

i) it is closed under the operation.
Let y₁,y₂∈Y , then f(y₁)f(y₂)=y₁,y₂
ii) it is closed under inverses.
Let y∈Y,???

4. Originally Posted by hzhang610
ii) it is closed under inverses.
Let y∈Y,???
Try this. Let $\sigma$ be a permutation leaving $y\in Y$ fixed. Since it is a bijection it means there is an inverse permutation, $\sigma^{-1}$. Argue this inverse permutation is what you are looking for as an inverse.