1.1.17 The Symmetric/Dihedral Isomorphism

We recall that the $S_{3}$ and di­he­dral group $D_{3}$ are iso­mor­phic by the De­f­i­n­i­tion (1.1.80), since these are the only group orders $(n = 3\ )$, in which the con­di­tion $n! = 2n$ is met.

De­f­i­n­i­tion (1.1.84) - For two groups $G$ and $G^{'}$ with de­fined binary op­er­a­tions $\circ$* and* $*$ re­spec­tive­ly, an Iso­mor­phism be­tween these groups exists if the map­ping $\varphi:G \rightarrow G'$, is a bi­jec­tion from $G \rightarrow G'$, and if $\varphi(x \circ y) = \varphi(x)*\varphi(y)$, for all $x,y \in G.$ Two iso­mor­phic groups are de­not­ed as $G \cong G^{'}.$

As the double-row per­mu­ta­tions of $S_{3}$ are in a one-to-one cor­re­spon­dence (bi­jec­tive) with the per­mu­ta­tion ma­tri­ces of $D_{3}$, the binary op­er­a­tion of mul­ti­pli­ca­tion is the same for both groups, im­ply­ing $S_{3} \cong D_{3}.$We show the bi­jec­tion be­tween the el­e­ments of each group below.