Symmetry Group of Equilateral Triangle/Cayley Table

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 $1.9$