Symmetry Group of Regular Pentagon/Cayley Table

Cayley Table of Symmetry Group of Regular Pentagon

 * Symmetry-Group-of-Regular-Pentagon.png

The Cayley table of the symmetry group of the regular pentagon can be written:


 * $\begin{array}{c|cccccc}

& e  & r   & r^2 & r^3 & r^4 & t_A & t_B & t_C & t_D & t_E \\ \hline e  & e   & r   & r^2 & r^3 & r^4 & t_A & t_B & t_C & t_D & t_E \\ r  & r   & r^2 & r^3 & r^4 & e   & t_C & t_D & t_E & t_A & t_B \\ r^2 & r^2 & r^3 & r^4 & e  & r   & t_E & t_A & t_B & t_C & t_D \\ r^3 & r^3 & r^4 & e  & r   & r^2 & t_B & t_C & t_D & t_E & t_A \\ r^4 & r^4 & e  & r   & r^2 & r^3 & t_D & t_E & t_A & t_B & t_C \\ t_A & t_A & t_D & t_B & t_E & t_C & e  & r^2 & r^4 & r   & r^3 \\ t_B & t_B & t_E & t_C & t_A & t_D & r^3 & e  & r^2 & r^4 & r   \\ t_C & t_C & t_A & t_D & t_B & t_E & r  & r^3 & e   & r^2 & r^4 \\ t_D & t_D & t_B & t_E & t_C & t_A & r^4 & r  & r^3 & e   & r^2 \\ t_E & t_E & t_C & t_A & t_D & t_B & r^2 & r^4 & r  & r^3 & e   \\ \end{array}$ where the various symmetry mappings of the regular pentagon $\PP = ABCDE$ are:
 * the identity mapping $e$
 * the rotations $r, r^2, r^3, r^4$ of $72^\circ, 144^\circ, 216^\circ, 288^\circ$ around the center of $\PP$ anticlockwise respectively
 * the reflections $t_A, t_B, t_C, t_D, t_E$ about the lines through the center of $\PP$ and the vertices $A$ to $E$ respectively.