Introduction to Groups

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 that

Describe in pictures or words the elements of $D_5$ (symmetries of a regular pentagon).

For $n \geq 3$, describe the elements of $D_n$. 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.

Suppose that an element $X$ of a dihedral group is the product of $m$ rotations and $n$ reflections. Complete the following statement: $X$ is a rotation if and only if .

Find elements $A$, $B$, and $C$ in $D_4$ such that $AB = BC$ but $A \neq C$. (Thus, ``cross cancellation'' is not valid.)

Explain what the following diagram proves about the group $D_n$: it shows the vertices $1, 2, \ldots , n$ of a regular $n$-gon labeled around the boundary, and depicts applying a reflection $F$ followed by the rotation $R_{360/n}$ to relabel the vertices, compared with applying $R_{360/n}$ followed by $F$ to the same starting configuration, so that both composite motions can be tracked vertex by vertex.

Describe the symmetries of a nonsquare rectangle. Construct the corresponding Cayley table. Is this group Abelian? What property does this group have in common with the dihedral groups?

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:

Describe the symmetries of this strip. Is the group of symmetries of the strip Abelian?

For each of the snowflakes in the figure (four photographed snow crystals, each with visible six-fold symmetry), 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 in the figure as a roughly circular arrangement of $28$ evenly spaced spikes around a central capsid.

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.

For each design below (twelve designs with varying rotational and reflective symmetry, including a biohazard symbol, a pentagon-with-star logo, and several floral and pinwheel patterns), determine the symmetry group (ignore imperfections).

Let $n$ be a positive integer greater than or equal to $3$. Prove that $D_n$ contains $R_{180}$ if and only if $n$ is even.

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 $X^3 = F$.