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==Examples==
==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.
The symmetric [[Group (Mathematics)|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]]
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>.
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>.
==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.
The symmetric group is non-abelian - that is, <math>a*b \neq b*a \ </math> in the symmetric group.
== Unsolved Problem ==
== Unsolved Problem ==
A proposition known as "Netto's conjecture" (proven by Dixon in 1969) states that the probability that two elements P<sub>1</sub> and P<sub>2</sub> 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.<ref>http://mathworld.wolfram.com/SymmetricGroup.html</ref>
A proposition known as "Netto's conjecture" (proven by Dixon in 1969) states that the probability that two elements P<sub>1</sub> and P<sub>2</sub> of a symmetric group can [[Group Generator|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.<ref>http://mathworld.wolfram.com/SymmetricGroup.html</ref>
==References==
{{Reflist}}