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 $BC$.


 * Symmetry-Group-of-Isosceles-Triangle.png

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

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

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

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

We also have that:

are also groups of order $2$.

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