1.1.18 The $Symmetric/Dihedral$ Group $\in 3\mathbb{R}$

We aim to study the geo­met­ric spaces nec­es­sary for groups to orient them­selves for their binary op­er­a­tions to occur. We fur­ther dis­tin­guish the $S_{3}/D_{3}$ group as re­sid­ing in the real geo­met­ric space$\ \in \mathbb{R}^{3}$, re­fer­ring to this group as $S_{3\mathbb{R}}/D_{3\mathbb{R}}$. We also note the nomen­cla­ture of the el­e­ments of the group to dis­tin­guish be­tween the cyclic sub­group, de­not­ed as $Y_{j}$, and the non-abelian sub­group, rep­re­sent­ed as $X_{j}$. The reason for this naming will become more ap­par­ent as we progress. For the sub­groups of el­e­ments $\left\{ Y_{1},Y_{2},Y_{3} \right\} \subset$ $S_{3\mathbb{R}}/D_{3\mathbb{R}}$, we have

and the el­e­ments that form the sub­group $\left\{ X_{1},X_{2},X_{3} \right\} \subset$ $S_{3\mathbb{R}}/D_{3\mathbb{R}}$, have the rep­re­sen­ta­tions

De­f­i­n­i­tion (1.1.88) – The Sym­met­ric/Di­he­dral Group $\mathbf{S}_{\mathbf{3}\mathbb{R}}\mathbf{/}\mathbf{D}_{\mathbf{3}\mathbb{R}}$ is a group under com­po­si­tion that sits in the com­plex geo­met­ric space $\mathbb{R}^{3 \times 3}$, with a set of per­mu­ta­tion el­e­ments $A^{\mathbb{R}} = \left\{ X_{j},Y_{j} \right\}$

The set of per­mu­ta­tion el­e­ments $A^{\mathbb{R}} \in S_{3\mathbb{R}}/D_{3\mathbb{R}},$have values $\in \mathbb{R}^{+}$, and the set of per­mu­ta­tions $A^{\mathbb{R}}$ forms a group under com­po­si­tion since it is con­sis­tent with the group axioms, as stated in the fol­low­ing the­o­rem.

The­o­rem (1.1.89) – The set of per­mu­ta­tions $A^{\mathbb{R}} \in S_{3\mathbb{R}}/D_{3\mathbb{R}}$ forms a group under com­po­si­tion

Proof – Ver­i­fi­ca­tion of group axioms

• Clo­sure: All $A^{\mathbb{R}}$ el­e­ments are bi­jec­tive, and the com­po­si­tion of any el­e­ments $\phi \circ \psi\ \in A^{\mathbb{R}}$ is also bi­jec­tive.

• As­so­cia­tiv­i­ty: Per­mu­ta­tions are func­tions, and their com­po­si­tions are as­so­cia­tive.

• Iden­ti­ty: The iden­ti­ty func­tion $\epsilon_{A^{\mathbb{R}}}$ bi­jec­tive­ly maps all set el­e­ments${\ A}^{\mathbb{R}}$ to them­selves $\left( \epsilon_{A^{\mathbb{R}}}:A^{\mathbb{R}} \rightarrow A^{\mathbb{R}} \right)$.

• In­verse: $\phi$ is a per­mu­ta­tion and thus bi­jec­tive, and a func­tion exists $\phi^{- 1}$ that is also bi­jec­tive.

$\square$

We are now fa­mil­iar with the group axioms of clo­sure, as­so­cia­tiv­i­ty, the iden­ti­ty, and the in­verse func­tion, which we will adhere to as re­quire­ments for the groups pre­sent­ed in this book. Sym­met­ric groups rep­re­sent a group con­tain­ing all the pos­si­ble per­mu­ta­tions, which is equal to $n!$for the sym­me­try group$\ S_{n}$. In­ter­est­ing­ly, the per­mu­ta­tions of the equi­lat­er­al tri­an­gle from the group's com­po­si­tion $D_{3}$, are iso­mor­phic to the sym­me­try group $S_{3}$.