Symmetric group on three points
= Symmetric group on three points
{title2=$S_3$}
= Symmetric group on three letters
{synonym}
The <permutation group> of a three-element set has six elements: the identity, three <transpositions>, and two three-cycles. A three-cycle and any transposition generate it, with relations $x^3=h^2=e$, $hxh^{-1}=x^{-1}$. Its <alternating group> is the normal cyclic subgroup of order three. It is the unique nonabelian isomorphism type of a <group of order six>.