# Isomorphisms between Symmetry Groups of Isosceles Triangle and Equilateral Triangle

## 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:

 $\ds$  $\ds \set {e, \tuple {12} }$ $\ds$  $\ds \set {e, \tuple {13} }$ $\ds$  $\ds \set {e, \tuple {23} }$

are also groups of order $2$.

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

$\blacksquare$