Chapter Questions
With pictures and words, describe each symmetry in $D_{3}$ (the set of symmetries of an equilateral triangle).
Write out a complete Cayley table for $D_{3}$. Is $D_{3}$ Abelian?
In $D_{4}$, find all elements $X$ such thata. $X^{3}=V$;b. $X^{3}=R_{90}$;c. $X^{3}=R_{0}$;d. $X^{2}=R_{0}$;e. $X^{2}=H$.
Describe in pictures or words the elements of $D_{5}$ (symmetries of a regular pentagon).
For $n \geq 3$, describe the elements of $D_{n}$. (Hint: You will need to consider two cases $-n$ even and $n$ odd.) How many elements does $D_{n}$ have?
In $D_{n}$, explain geometrically why a reflection followed by a reflection must be a rotation.
In $D_{n}$, explain geometrically why a rotation followed by a rotation must be a rotation.
In $D_{n}$, explain geometrically why a rotation and a reflection taken together in either order must be a reflection.
Associate the number 1 with a rotation and the number $-1$ with a reflection. Describe an analogy between multiplying these two numbers and multiplying elements of $D_{n}$.
If $r_{1}, r_{2}$, and $r_{3}$ represent rotations from $D_{n}$ and $f_{1}, f_{2}$, and $f_{3}$ represent reflections from $D_{n}$, determine whether $r_{1} r_{2} f_{1} r_{3} f_{2} f_{3} r_{3}$ is a rotation or a reflection.
Suppose that $a, b$, and $c$ are elements of a dihedral group. Is $a^{2} b^{4} a c^{5} a^{3} c$ a rotation or a reflection? Explain your reasoning.
Which letters of the alphabet written in upper case block style have a symmetry group with four elements? Describe the four symmetries.
Find elements $A, B$, and $C$ in $D_{4}$ such that $A B=B C$ but $A \neq C$. (Thus, "cross cancellation" is not valid.)
Explain what the following diagram proves about the group $D_{n}$,
Describe the symmetries of a nonsquare rectangle. Construct the corresponding Cayley table.
Describe the symmetries of a parallelogram that is neither a rectangle nor a rhombus. Describe the symmetries of a rhombus that is not a rectangle.
Describe the symmetries of a noncircular ellipse. Do the same for a hyperbola.
Consider an infinitely long strip of equally spaced H's:$\cdots \mathrm{H} \mathrm{H} \mathrm{H} \mathrm{H} \cdots$Describe the symmetries of this strip. Is the group of symmetries of the strip Abelian?
For each of the snowflakes in the figure, find the symmetry group and locate the axes of reflective symmetry (disregard imperfections).
Determine the symmetry group of the outer shell of the cross section of the human immunodeficiency virus (HIV) shown below.
Let $X, Y, R_{90}$ be elements of $D_{4}$ with $Y \neq R_{90}$ and $X^{2} Y=R_{90}$. Determine $Y$. Show your reasoning.
If $F$ is a reflection in the dihedral group $D_{n}$ find all elements $X$ in $D_{n}$ such that $X^{2}=F$ and all elements $X$ in $D_{n}$ such that $X^{3}=F$.
What symmetry property do the words "mow," "sis," and "swims" have when written in uppercase letters?
For each design below, determine the symmetry group (ignore imperfections).
What group theoretic property do uppercase letters $\mathrm{F}, \mathrm{G}, \mathrm{J}, \mathrm{L}, \mathrm{P}, \mathrm{Q}, \mathrm{R}$ have that is not shared by the remaining uppercase letters in the alphabet?