Symmetry Group of Equilateral Triangle is Symmetric Group

Theorem

Let $D_3$ denote the symmetry group of the equilateral triangle.

Let $S_3$ denote the symmetric group on $3$ letters.


Then $D_3$ is isomorphic to $S_3$.


Proof


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Follows from Symmetric Group on 3 Letters is Isomorphic to Dihedral Group D3.

$\blacksquare$

Sources

  • 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Examples of Group Structure: $\S 30 \gamma$
  • 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(iv)}$
  • 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.19$
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