Isomorphisms between Symmetry Groups of Isosceles Triangle and Equilateral Triangle

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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 $S_3$:

\(\ds \) \(\) \(\ds \set {e, \tuple {12} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {13} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {23} }\)


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.