Identity/Examples/Symmetry Group of Square
Jump to navigation
Jump to search
Example of Identity Element
Consider the symmetry group of the square:
Let $\SS = ABCD$ be a square.
The various symmetry mappings of $\SS$ 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 in the $x$ and $y$ axis respectively
- the reflection $t_{AC}$ in the diagonal through vertices $A$ and $C$
- the reflection $t_{BD}$ in the diagonal through vertices $B$ and $D$.
This group is known as the symmetry group of the square, and can be denoted $D_4$.
The mapping $e$ which leaves $\SS$ unchanged is the identity element.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses