The Symmetric Group of Degree $n$ Is A Group

We've already shown this in our work earlier in the semester. Given any two permutations of $X$, their composition is a permutation of $X$ so $S_n$ is closed. We also know that function composition is associative. The identity permutation $\text{id}_X$ definded by $\text{id}_X(x)=x$ is an element of $S_n$. The inverse of $\alpha \in S_n$ is the inverse function $\alpha^{-1}$, which is again a permutation and so in $S_n$. This shows that $S_n$ satisfies the definition of being a group.