Stabilizer in Group of Transformations

Theorem
Let $$X$$ be any set with $$n$$ elements (where $$n \in \N^*$$).

Consider the Symmetric Group $$S_n$$ as a group of transformations on $$X$$.

Let $$x \in X$$.

Then the stabilizer of $$x$$ is isomorphic to $$S_{n-1}$$.

Proof
Consider the subset of natural numbers $$\N^*_n = \left\{{1, 2, \ldots, n}\right\}$$.

By the definition of cardinality, $$H$$ is equivalent to $$\N^*_n$$ and WLOG we can consider $$S_n$$ acting directly on $$\N^*_n$$.

The stabilizer of $$n$$ in $$\N^*_n$$ is all the permutations of $$S_n$$ which fix $$n$$, which is clearly $$S_{n-1}$$.

A permutation can be applied to $$\N^*_n$$ so that $$i \to n$$ for any $$i$$.

Thus one can build an isomorphism to show the result for a general $$i$$.