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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, there exist <math>a,b</math> such that <math>a*b \neq b*a \ </math> in the symmetric group.

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>.

== Unsolved Problem ==

== Unsolved Problem ==