1.1.14 Symmetric and Dihedral Groups

The study of the sym­met­ric groups is im­por­tant in group theory, par­tic­u­lar­ly the uni­tary groups that are part of the matrix group conics that we will ul­ti­mate­ly build upon. Per­mu­ta­tions were one of the ear­li­est rep­re­sen­ta­tions of sym­me­tries in group theory, and these were a means to show how an object or set could be re­arranged. We define a per­mu­ta­tion of a set that forms a group under com­po­si­tion, as

De­f­i­n­i­tion (1.1.75)$\ -$ A per­mu­ta­tion of a set $A$ is a bi­jec­tion $\phi:A \rightarrow A$.

The­o­rem (1.1.76) – For $A$ as a set, the set of per­mu­ta­tions of $S_{X}$ forms a group under com­po­si­tion.

Proof – Ver­i­fi­ca­tion of group axioms

• Clo­sure: If $\phi$ and $\psi$ are bi­jec­tive, then the com­po­si­tion of $\phi \circ \psi$ is as well.

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

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

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

$\square$

The Irish math­e­mati­cian Arthur Cayley showed just how im­por­tant the sym­met­ric groups are by prov­ing that every group is a sub­group of some sym­met­ric group¹.

The­o­rem (1.1.77) – Cayley's The­o­rem. Every group $G$ is iso­mor­phic to a group of per­mu­ta­tions.

Proof $-$For each el­e­ment $x \in G$, con­sid­er the func­tion $f_{x}:G \rightarrow$ to rep­re­sent left mul­ti­pli­ca­tion by $x$ on $g$, $f_{x}:g \mapsto xg.$ There are two claims im­plied in the the­o­rem:

• Each $f_{x}$ is a per­mu­ta­tion of $G.$

• A group iso­mor­phic to $G$ is formed by $\left\{ f_{x}:x \in G, \circ \right\}.$

For the first claim, taking $f_{x^{- 1}}$ as the in­verse func­tion of $f_{x}$:

and as such, each func­tion $f_{x}$ is a bi­jec­tion. For the second claim, we define the map­ping $\phi:G \rightarrow \left\{ f_{x} \right\}$, by $\phi(x) = f_{x}$ to be an iso­mor­phism, for the one-to-one re­la­tion

where every per­mu­ta­tion func­tion $f_{x}$, is the image of $\phi(x)$ and is onto. In­jec­tive ho­mo­mor­phism is sat­is­fied for $x,y \in G$, there­fore

from which we have

$\square$

1. Cayley, Arthur (1854), "On the theory of groups as de­pend­ing on the sym­bol­ic equa­tion $\theta^{n}$ = 1", Philo­soph­i­cal Mag­a­zine, 7 (42): 40–47