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 $8.6$