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.
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1$\begingroup$ See if this helps: en.wikipedia.org/wiki/… $\endgroup$– JC574Aug 5, 2014 at 15:07
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3 Answers
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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 $\mathfrak S_n$.
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$\begingroup$ Thanks a lot. Goes $\mathrm{Aut}$ has some meaning? $\endgroup$ Aug 5, 2014 at 15:13
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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
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$\begingroup$ @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. $\endgroup$– user4894Aug 5, 2014 at 15:49
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$\begingroup$ Is there an accepted notation for the set of permutation matrices for a given value of n? $\endgroup$– LoriJul 2, 2021 at 17:03
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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 $\mathcal S_X$.
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In addition to the answers above, it can also be denoted by
$$X!$$
This notation has the neat property that
$$|X!| = |X|!$$
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$\begingroup$ 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. $\endgroup$– BenApr 14 at 2:54