How many isomorphisms from a set to itself

Question

I am reading through Spivak's "Category Theory for Scientists" and one of the exercises is to find the number of Isomorphisms from a set X to itself. My attempt: If I am not mistaken, if |X|= n where n∈ℕ then |Hom(X,X)| is the square of |X| but I have no idea how to extrapolate the number of Isomorphisms in Hom(X,X). Any help would be greatly appreciated.

Answer 1

An isomorphism from a set to itself is an automorphism.

An automorphism is a permutation of the set since it is one-to-one and onto. Assuming the set $X$ is finite, ie. has $n$ elements, then the problem is reduced to how many ways can you rearrange $n$ elements, so the answer is $n!$.

Answer 2

I'm assuming the question is: In the category $\mathbf{FinSet}$ what is the cardinality of $\DeclareMathOperator{Iso}{Iso}\Iso(X,X)$?

We first have to understand what the isomorphisms are in $\mathbf{FinSet}$. It isn't too difficult to see these are exactly the bijections. So the cardinality of $\Iso(X,X)$ is exactly the number of bijections $X\to X$. This number is exactly $n!$ where $n=\lvert X\rvert$.