Definition:Symmetry Group of Regular Pentagon

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Group Example

Let $\PP = ABCDE$ be a regular pentagon.

Symmetry-Group-of-Regular-Pentagon.png

The various symmetry mappings of $\PP$ 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 $t_A, t_B, t_C, t_D, t_E$ in the lines through the center of $\PP$ and the vertices $A$ to $E$ respectively.

This group is known as the symmetry group of the regular pentagon.


Cayley Table

Symmetry-Group-of-Regular-Pentagon.png

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

$\begin {array} {c|cccccc} \circ & e & r & r^2 & r^3 & r^4 & t_A & t_B & t_C & t_D & t_E \\ \hline e & e & r & r^2 & r^3 & r^4 & t_A & t_B & t_C & t_D & t_E \\ r & r & r^2 & r^3 & r^4 & e & t_C & t_D & t_E & t_A & t_B \\ r^2 & r^2 & r^3 & r^4 & e & r & t_E & t_A & t_B & t_C & t_D \\ r^3 & r^3 & r^4 & e & r & r^2 & t_B & t_C & t_D & t_E & t_A \\ r^4 & r^4 & e & r & r^2 & r^3 & t_D & t_E & t_A & t_B & t_C \\ t_A & t_A & t_D & t_B & t_E & t_C & e & r^2 & r^4 & r & r^3 \\ t_B & t_B & t_E & t_C & t_A & t_D & r^3 & e & r^2 & r^4 & r \\ t_C & t_C & t_A & t_D & t_B & t_E & r & r^3 & e & r^2 & r^4 \\ t_D & t_D & t_B & t_E & t_C & t_A & r^4 & r & r^3 & e & r^2 \\ t_E & t_E & t_C & t_A & t_D & t_B & r^2 & r^4 & r & r^3 & e \\ \end {array}$


Also known as

The symmetry group of the regular pentagon is also known as:

the dihedral group of order $10$ and denoted $D_5$


Some sources denote $D_5$ as ${D_5}^*$.


Also see

  • Results about Symmetry Group of Regular Pentagon can be found here.


Sources