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.


3 Answers 3


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 $\mathfrak S_n$.

  • $\begingroup$ Thanks a lot. Goes $\mathrm{Aut}$ has some meaning? $\endgroup$ Aug 5, 2014 at 15:13
  • 3
    $\begingroup$ @user3761729 the notation makes sense in the context of category theory. The bijection of a set A to itself are the automorphism of A in the category Set. See en.wikipedia.org/wiki/Category_of_sets $\endgroup$
    – quid
    Aug 5, 2014 at 15:16
  • $\begingroup$ @user3761729 Aut = automorphism group. Set of bijections of the set, under composition. A permutation can be thought of as a bijection from the set to itself. $\endgroup$
    – user4894
    Aug 5, 2014 at 15:49
  • $\begingroup$ Does this change if $X$ is a multiset? $\endgroup$
    – FUZxxl
    Jun 11, 2017 at 0:35
  • $\begingroup$ Is there an accepted notation for the set of permutation matrices for a given value of n? $\endgroup$
    – Lori
    Jul 2, 2021 at 17:03

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


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


This notation has the neat property that

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

  • $\begingroup$ Interesting (+1), but not sure I like this notation --- the dual use of $!$ to denote operations on two different objects seems as likely to invite confusion as clarity. $\endgroup$
    – Ben
    Apr 14 at 2:54

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