Symmetric Group on 3 Letters/Group Presentation

Group Presentation of Symmetric Group on $3$ Letters

The group presentation of the symmetric group on $3$ letters is given by:

$S_3 := \gen {a, b: a^3 = b^2 = \paren {a b}^2 = e}$

Hence:

$\begin{array}{c|cccccc} & e & a & a^2 & b & a b & a^2 b \\ \hline e & e & a & a^2 & b & a b & a^2 b \\ a & a & a^2 & e & a b & a^2 b & b \\ a^2 & a^2 & e & a & a^2 b & b & a b \\ b & b & a^2 b & a b & e & a^2 & a \\ a b & a b & b & a^2 b & a & e & a^2 \\ a^2 b & a^2 b & a b & b & a^2 & a & e \\ \end{array}$


Proof

Let $G = \gen {a, b: a^3 = b^2 = \paren {a b}^2 = e}$.

It is to be demonstrated that $S_3$ is isomorphic to $G$.

Consider the Cayley table for $S_3$:

$\begin{array}{c|cccccc}\circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$


We demonstrate that $S_3$ is isomorphic to $G$ as follows:

Let $\phi: G \to S_3$ be a mapping that sends:

$\phi: a \mapsto p$
$\phi: b \mapsto r$

We have:

$p^3 = e$
$r^2 = e$
$\paren {p r}^2 = s^2 = e$

demonstrating that $S_3$ has the same group presentation as $G$.

Hence the result.

$\blacksquare$


Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem
  • 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): $1$: Subgroups: Problem $1.1$
  • 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Example $5.2$
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