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 $\mathbb {S} _{n}\$ and has $n!\$. 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 $\mathbb {S} _{4}\$, and it is written $(1432)\$, because the first point went to the fourth position (ie, $1\rightarrow 4\$), the fourth point went to the third position (ie, $4\rightarrow 3\$), 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 $(1327)\$ and $(2534)\$ in $\mathbb {S} _{7}\$ would be computed as so:

$ABCDEFG\$

becomes, under $(1327)\$,

$GCADEFB\$,

and once we perform $(2534)\$ on this string, it becomes

$GDEACFB\$.

In this final arrangement, A has ended up at position 4 (ie, $1\rightarrow 4\$), D has ended up at position 2, and so on, until we discover the product of the two permutations to be $(1427)(35)\$.

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 terms in the sequence of a symmetric group.