Symmetry Group of Square/Cayley Table

From ProofWiki
Jump to navigation Jump to search

Cayley Table of Symmetry Group of Square

SymmetryGroupSquare.png


The Cayley table of the symmetry group of the square can be written:

$\begin{array}{c|cccccc}
      & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\

\hline e & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\ r & r & r^2 & r^3 & e & t_{AC} & t_{BD} & t_y & t_x \\ r^2 & r^2 & r^3 & e & r & t_y & t_x & t_{BD} & t_{AC} \\ r^3 & r^3 & e & r & r^2 & t_{BD} & t_{AC} & t_x & t_y \\ t_x & t_x & t_{BD} & t_y & t_{AC} & e & r^2 & r^3 & r \\ t_y & t_y & t_{AC} & t_x & t_{BD} & r^2 & e & r & r^3 \\ t_{AC} & t_{AC} & t_x & t_{BD} & t_y & r & r^3 & e & r^2 \\ t_{BD} & t_{BD} & t_y & t_{AC} & t_x & r^3 & r & r^2 & e\\ \end{array}$

where the various symmetry mappings of the square $\SS = ABCD$ are:

the identity mapping $e$
the rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ around the center of $\SS$ anticlockwise respectively
the reflections $t_x$ and $t_y$ are reflections about the $x$ and $y$ axis respectively
the reflection $t_{AC}$ about the diagonal through vertices $A$ and $C$
the reflection $t_{BD}$ about the diagonal through vertices $B$ and $D$.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Symmetry_Group_of_Square/Cayley_Table&oldid=545401"