# Changes

The symmetric [[Group (Mathematics)|group]] on a set of N points is often written $\mathbb{S}_n \$ and has $n! \$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 $\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.
The symmetric group contains several subgroups: notably, $\mathbb{S}_n \$ contains every symmetric group $\mathbb{S}_m \$ as a subgroup so long as $m \leq n \$. The symmetric group also contains as a subgroup the alternating group $\mathbb{A}_n \$, which consists only of even permutations on n points.
The symmetric group is non-abelian - that is, there exist $a,b$ such that $a*b \neq b*a \$ in the symmetric group.