# symmetric group

Let $X$ be a set. Let $S(X)$ be the set of permutations of $X$ (i.e. the set of bijective functions on $X$). Then the act of taking the composition of two permutations induces a group structure on $S(X)$. We call this group the and it is often denoted ${\rm Sym}(X)$.

When $X$ has a finite number $n$ of elements, we often refer to the symmetric group as $S_{n}$, and describe the elements by using cycle notation.

Title symmetric group SymmetricGroup1 2013-03-22 14:03:53 2013-03-22 14:03:53 antizeus (11) antizeus (11) 5 antizeus (11) Definition msc 20B30 Symmetry2