Symmetric Group on 3 Letters is Isomorphic to Dihedral Group D3
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Theorem
Let $S_3$ denote the Symmetric Group on 3 Letters.
Let $D_3$ denote the dihedral group $D_3$.
Then $S_3$ is isomorphic to $D_3$.
Proof
Consider $S_3$ as presented by its Cayley table:
- $\begin{array}{c|cccccc}\circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$
Consider $D_3$ as presented by its group presentation:
- $D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$
and its Cayley table:
- $\begin{array}{c|cccccc}
& e & a & a^2 & b & a b & a^2 b \\
\hline e & e & a & a^2 & b & a b & a^2 b \\ a & a & a^2 & e & a b & a^2 b & b \\ a^2 & a^2 & e & a & a^2 b & b & a b \\ b & b & a^2 b & a b & e & a^2 & a \\ a b & a b & b & a^2 b & a & e & a^2 \\ a^2 b & a^2 b & a b & b & a^2 & a & e \\ \end{array}$
Let $\phi: S_3 \to D_3$ be specified as:
\(\ds \map \phi {1 2 3}\) | \(=\) | \(\ds a\) | ||||||||||||
\(\ds \map \phi {2 3}\) | \(=\) | \(\ds b\) |
Then by inspection, we see:
\(\ds \map \phi {1 3 2}\) | \(=\) | \(\ds a^2\) | ||||||||||||
\(\ds \map \phi {1 3}\) | \(=\) | \(\ds a b\) | ||||||||||||
\(\ds \map \phi {1 2}\) | \(=\) | \(\ds a^2 b\) |
and the result follows.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Exercise $8.6$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Example $8.3$