Notation for all permutations of a set

Question

Suppose I have a finite set $X$. Is there a standard notation to denote the set of all possible permutations of the elements of $X$?

P.S. something like the power set notation for all subsets.

Answer 1

The group of the permutations of $X$ (even if $X$ is infinite) is denoted by : $S(X)$, $\mathrm{Aut}(X)$, or $\mathfrak{S}(X)$.

If $X$ is finite with $n$ elements, it is denoted by $S_n$ or $\mathfrak S_n$.

Answer 2

I think you are looking for the symmetric group for which there are several notations, e.g. $\mathfrak{G}_X$ or $\mathcal S_X$.

Answer 3

In addition to the answers above, it can also be denoted by

$$X!$$

This notation has the neat property that

$$|X!| = |X|!$$