# Isomorphisms between Symmetry Groups of Isosceles Triangle and Equilateral Triangle

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## Theorem

Let $\TT = ABC$ be an isosceles triangle whose apex is $A$.

Let $\struct {\TT, \circ}$ be the symmetry group of $\TT$, where the symmetry mappings are identified as:

- the identity mapping $e$
- the reflection $d$ in the line through $A$ and the midpoint of $BC$.

Let $\SS = A'B'C'$ be an equilateral triangle.

We define in cycle notation the following symmetry mappings on $\triangle A'B'C'$:

\(\ds e\) | \(:\) | \(\ds \tuple {A'} \tuple {B'} \tuple {C'}\) | Identity mapping | |||||||||||

\(\ds p\) | \(:\) | \(\ds \tuple {A'B'C'}\) | Rotation of $120 \degrees$ anticlockwise about center | |||||||||||

\(\ds q\) | \(:\) | \(\ds \tuple {A'C'B'}\) | Rotation of $120 \degrees$ clockwise about center | |||||||||||

\(\ds r\) | \(:\) | \(\ds \tuple {B'C'}\) | Reflection in line $r$ | |||||||||||

\(\ds s\) | \(:\) | \(\ds \tuple {A'C'}\) | Reflection in line $s$ | |||||||||||

\(\ds t\) | \(:\) | \(\ds \tuple {A'B'}\) | Reflection in line $t$ |

Then $\struct {\TT, \circ}$ is isomorphic to the $3$ subgroups of $S_3$:

\(\ds \) | \(\) | \(\ds \set {e, \tuple {12} }\) | ||||||||||||

\(\ds \) | \(\) | \(\ds \set {e, \tuple {13} }\) | ||||||||||||

\(\ds \) | \(\) | \(\ds \set {e, \tuple {23} }\) |

## Proof

We have that $\struct {\TT, \circ}$ is of order $2$.

We also have that:

\(\ds \) | \(\) | \(\ds \set {e, \tuple {12} }\) | ||||||||||||

\(\ds \) | \(\) | \(\ds \set {e, \tuple {13} }\) | ||||||||||||

\(\ds \) | \(\) | \(\ds \set {e, \tuple {23} }\) |

From Parity Group is Only Group with 2 Elements, all these groups are isomorphic.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Exercise $8.6$