Thread: 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

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.

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,???

Originally Posted by hzhang610

ii) it is closed under inverses.
Let y∈Y,???

Try this. Let $\displaystyle \sigma$ be a permutation leaving $\displaystyle y\in Y$ fixed. Since it is a bijection it means there is an inverse permutation, $\displaystyle \sigma^{-1}$. Argue this inverse permutation is what you are looking for as an inverse.