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.
The symmetric group on a set of N points is often written and has 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 , and it is written , because the first point went to the fourth position (i.e., ), the fourth point went to the third position (i.e., ), the third point went to the second position, and the second point went to the first position.
The operation of the symmetric group is re-arrangement composition: for example, the "product" of and in would be computed as so:
becomes, under ,
and once we perform on this string, it becomes
In this final arrangement, A has ended up at position 4 (i.e., ), D has ended up at position 2, and so on, until we discover the product of the two permutations to be .
The symmetric group contains several subgroups: notably, contains every symmetric group as a subgroup so long as . The symmetric group also contains as a subgroup the alternating group , which consists only of even permutations on n points.
For the symmetric group is non-abelian - that is, there exist such that in the symmetric group. is the trivial group and is isomorphic to .
The order of is
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.