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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.

user3761729
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    See if this helps: http://en.wikipedia.org/wiki/Symmetric_group#Definition_and_first_properties – JC574 Aug 05 '14 at 15:07

3 Answers3

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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 $\mathfrak S_n$.

D.L.
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  • Thanks a lot. Goes $\mathrm{Aut}$ has some meaning? – user3761729 Aug 05 '14 at 15:13
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    @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 http://en.wikipedia.org/wiki/Category_of_sets – quid Aug 05 '14 at 15:16
  • @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. – user4894 Aug 05 '14 at 15:49
  • Does this change if $X$ is a multiset? – FUZxxl Jun 11 '17 at 00:35
  • Is there an accepted notation for the set of permutation matrices for a given value of n? – Lori Jul 02 '21 at 17:03
  • $\operatorname{Sym}(X)$ for the group of permutations of $X$ is also used. – Matthew Towers Dec 22 '22 at 09:26
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I think you are looking for the symmetric group for which there are several notations, e.g. $\mathfrak{G}_X$ or $\mathcal S_X$.

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

$$X!$$

This notation has the neat property that

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

user76284
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  • 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. – Ben Apr 14 '23 at 02:54