Symmetry Group of Equilateral Triangle/Cayley Table
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Cayley Table of Symmetry Group of Equilateral Triangle
The Cayley table of the symmetry group of the equilateral triangle can be written:
- $\begin{array}{c|ccc|ccc}
\circ & e & p & q & r & s & t \\ \hline e & e & p & q & r & s & t \\ p & p & q & e & s & t & r \\ q & q & e & p & t & r & s \\ \hline r & r & t & s & e & q & p \\ s & s & r & t & p & e & q \\ t & t & s & r & q & p & e \\ \end{array}$
where:
| \(\ds e\) | \(:\) | \(\ds (A) (B) (C)\) | Identity mapping | |||||||||||
| \(\ds p\) | \(:\) | \(\ds (ABC)\) | Rotation of $120 \degrees$ anticlockwise about center | |||||||||||
| \(\ds q\) | \(:\) | \(\ds (ACB)\) | Rotation of $120 \degrees$ clockwise about center | |||||||||||
| \(\ds r\) | \(:\) | \(\ds (BC)\) | Reflection in line $r$ | |||||||||||
| \(\ds s\) | \(:\) | \(\ds (AC)\) | Reflection in line $s$ | |||||||||||
| \(\ds t\) | \(:\) | \(\ds (AB)\) | Reflection in line $t$ |
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.11$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.4$: Exercise $2.1$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(5)$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.9$