Symmetric group

A symmetric group is, simply stated, the collection of all permutations of a set. This concept has many applications in group theory of mathematics, along with one of its great unsolved problems.

Examples

The symmetric group on a set of N points is often written <math>\mathbb{S}_n \ </math> and has <math>n! \ </math> elements. 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 (i.e., <math>1 \rightarrow 4 \ </math>), the fourth point went to the third position (i.e., <math>4 \rightarrow 3 \ </math>), the third point went to the second position, and the second point went to the first position.

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 (i.e., <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>.

Properties

The symmetric group contains several subgroups: notably, <math>\mathbb{S}_n \ </math> contains every symmetric group <math>\mathbb{S}_m \ </math> as a subgroup so long as <math>m \leq n \ </math>. The symmetric group also contains as a subgroup the alternating group <math>\mathbb{A}_n \ </math>, which consists only of even permutations on n points.

For <math>n \geq 3 </math> the symmetric group is non-abelian - that is, there exist <math>a,b</math> such that <math>a*b \neq b*a \ </math> in the symmetric group. <math>\mathbb{S}_1 \ </math> is the trivial group and <math>\mathbb{S}_2 \ </math> is isomorphic to <math>\mathbb{Z}_2 \ </math>.

The order of <math>\mathbb{S}_n </math> is <math>n! </math>

Unsolved Problem

A proposition known as "Netto's conjecture" (proven by Dixon in 1969) states that the probability that two elements P₁ and P₂ of a symmetric group can generate the entire group approaches 3/4 as n increases to infinity. But a prominent unsolved problem in group theory is to find a general formula for the probability that two randomly selected elements generating the symmetric group on n points.