Changes

==Examples==

==Examples==

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.

The symmetric [[Group (Mathematics)|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 (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]]

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 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.

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.

== Unsolved Problem ==

== Unsolved Problem ==