1.1.15 The $Symmetric$ Groups

We can now define the sym­met­ric group $S_{n}$, as fol­lows:

De­f­i­n­i­tion (1.1.78) – The Sym­met­ric Group $\mathbf{S}_{\mathbf{n}}$is a group under com­po­si­tion, with a set of bi­jec­tive per­mu­ta­tion el­e­ments $A$

By com­bi­na­torics, the $S_{A}$ group will have $A!$el­e­ments $x = \left\{ 1,2,3,\ldots,X \right\}$, and if $X \geq 3$, then $S_{A}$ is non-abelian. We will use $S_{3}$ as an ex­am­ple, noting that any per­mu­ta­tion $\phi$ is both an in­jec­tive and sur­jec­tive func­tion (bi­jec­tive) from the set of three in­te­gers $\{ 1,2,3\}$, onto itself. The $S_{3}$ sym­me­try group rep­re­sents the ways to per­mute the set of three in­te­gers as men­tioned before, and one way to rep­re­sent its set of $3! = 6$ el­e­ments, is as fol­lows

where $\epsilon$ is the iden­ti­ty per­mu­ta­tion, $x,\ xy,\ xy^{2}$ rep­re­sent the three two-number swaps, and $y,y^{2}$ are cyclic per­mu­ta­tions in the left and right di­rec­tions. Keep­ing Cayley's The­o­rem in mind con­cern­ing the sym­met­ric groups, we can see that a proper un­der­stand­ing of them yields a deep un­der­stand­ing of all groups. As we pro­ceed, we will find the basis of the matrix groups dis­cussed are rooted in these sym­met­ric groups.