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A '''symmetric group''' is, simply stated, the collection of all permutations of a set. This concept has many applications in [[group theory]] of mathematics.
A '''symmetric group''' is, simply stated, the collection of all permutations of a set. This concept has many applications in [[group theory]] of mathematics.
==Examples==
The symmetric group on a set of N points is often written <math>\mathbb{S}_n \ </math> and has <math>n! \ </math>. Each element of a symmetric group is a way of re-arranging the points: for example, it is possible to re-arrange the points ABCD into so they read BCDA - this is an element of <math>\mathbb{S}_4 \ </math>, and it is written <math>(1432) \ </math>, because the first point went to the fourth position (ie, <math>1 \rightarrow 4 \ </math>), the fourth point went to the third position (ie, <math>4 \rightarrow 3 \ </math>), the third point went to the second position, and the second point went to the first position.
[[Category:mathematics]]
[[Category:mathematics]]
==Group Structure==
The operation of the symmetric group is re-arrangement composition: for example, the "product" of <math>(1327) \ </math> and <math>(2534) \ </math> in <math>\mathbb{S}_7 \ </math> would be computed as so:
<math>A B C D E F G \ </math>
becomes, under <math>(1327) \ </math>,
<math>G C A D E F B \ </math>,
and once we perform <math>(2534) \ </math> on this string, it becomes
<math>G D E A C F B \ </math>.
In this final arrangement, A has ended up at position 4 (ie, <math>1 \rightarrow 4 \ </math>), D has ended up at position 2, and so on, until we discover the product of the two permutations to be <math>(1427)(35) \ </math>.