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Topic 4 Dihedral groups: Discovery guide

Let's go back to groups of rotational symmetries.

Discovery 4.1.

Consider a thin, flat, solid in space with an equilateral triangle face.

A thin, flat, solid in space with an equilateral triangle face.
Figure 4.0.1. A thin, flat, solid in space with an equilateral triangle face.

This solid has six rotational symmetries. The group formed by these six rotations (under the operation of composition) is an example of a dihedral group. This particular dihedral group is denoted \(D_3\text{.}\)

Here are three of the six elements of \(D_3\text{:}\)

  • the trivial rotation of \(0\) radians;

  • rotation about the \(z\)-axis by \(2 \pi / 3\) radians counter-clockwise (when looking “down” the \(z\)-axis from “above”);

  • rotation about the \(x\)-axis by \(\pi\) radians.

As the trivial rotation acts as the group identity, denote it by \(e\) as usual. Of the other two rotations described above, it is traditional to denote the one about the \(z\)-axis by \(r\) and the one about the \(x\)-axis by \(s\text{.}\)

(a)

Describe each of the other three rotational symmetries of this shape in terms of an axis and angle of rotation.

(b)

Determine algebraic formulas for the three rotations from Task a as compositions involving \(r\) and \(s\text{.}\) (Careful: Remember that these are compositions of functions, so the right-most rotation is the first to be applied.)

(c)

Fill in the multiplication table for \(D_3\text{.}\)

Start by completing the reference row and column using your formulas from Task b. (Note that one of the reference blanks to fill in has been placed in between \(r\) and \(s\) — I hope it is obvious which formula should be placed there.)

The multiplication results in the rest of the table should all be expressed as either a single \(e\text{,}\) \(r\text{,}\) or \(s\text{,}\) or as one of your formulas in \(r\) and \(s\) from Task b. Use geometric analysis (perhaps by numbering or colour-coding the vertices and keeping track of where the end up?) to determine how each multiplication result should be expressed. Remember that products should be computed in the order left-hand-reference-column-entry-times-top-reference-row-entry, but that in a product of transformations the factor transformations are applied to the triangular plate right-to-left.

\(\times\) e r ? s ? ?
e
r
?
s
?
?
Figure 4.0.2.

(d)

Does order of multiplication matter in \(D_3\text{?}\) In particular, use your table to compare the results of \(r s\) and \(s r\) — are they the same?

(e)

Use your multiplication table to determine \(\inv{r}\) and \(\inv{s}\text{.}\) Then use algebra and the pattern of Task 2.10.b to determine the inverses of each of the other elements of \(D_3\text{,}\) and check your table to make sure your results are correct.

(f)

In carrying out algebraic calculations with the elements in \(D_3\text{,}\) it would be desirable to simplify all results to one of the six expressions in the reference row/column of the multiplication table. What algebra patterns did you discover while completing the multiplication table that might help in simplifying more complicated expressions? For example, hopefully you noticed that

\begin{equation*} s^2 = e \text{.} \end{equation*}

As another example, your inversion formulas from Task 4.1.e would be useful. What other algebraic simplification formulas can you find?

(g)

Are there any other non-algebraic patterns you notice in the multiplication table for \(D_3\text{?}\)

For an element \(g\) in a (multiplicative) group \(G\text{,}\) we call the smallest positive integer \(n\) for which

\begin{equation*} g^n = e \end{equation*}

the order of element \(g\) (if such an integer exists). In an additive group, the order of \(g\) is the smallest positive \(n\) for which

\begin{equation*} n \cdot g = 0 \end{equation*}

(again, if such an \(n\) exists).

Discovery 4.2.

Use your multiplication table to determine the order of each of the elements in \(D_3\text{.}\)

Discovery 4.3.

Consider a thin, flat, rectangular prism with a square face in space.

A thin, flat, rectangular prism with a square face in space.
Figure 4.0.3. A thin, flat, rectangular prism with a square face in space.

The group formed by the eight rotational symmetries (again, under the operation of composition) is another example of a dihedral group. This particular dihedral group is denoted \(D_4\text{.}\)

(a)

Based on your experience with \(D_3\text{,}\) which rotation of this square plate should be denoted \(r\text{,}\) and which should be denoted \(s\text{?}\)

Discovery 4.4.

In general, a flat, thin, solid whose face is a regular \(n\)-agon has a dihedral rotational symmetry group, \(D_n\text{.}\) Consider the algebraic patterns that \(D_3\) and \(D_4\) had in common, and conjecture how these patterns might generalize to \(D_n\text{.}\)

Discovery 4.5.

Can you come up with an example of an element in a group that has infinite order? That is, where the is no positive integer \(n\) so that \(g^n = e\) (or \(n \cdot g = 0\) if your example is an additive group)?

Discovery 4.6.

This activity will lead you through a proof that every element in a finite group must have finite order.

(a)

Suppose \(G\) is a finite group (i.e. contains a finite number of elements) and \(g\) is an element in \(G\text{.}\) Convince yourself that the list

\begin{equation*} e, g, g^2, g^3, \dotsc \end{equation*}

cannot contain an infinite number of different results.

(b)

Suppose \(k\) and \(m\) are positive integers so that

\begin{equation*} g^k = g^m \end{equation*}

but \(k \neq m\text{.}\) (Remember, you have already convinced yourself in Task a that the list of powers of \(g\) must contain some duplicates.) Use some group algebra to turn \(g^k = g^m \) into

\begin{equation*} g^? = e \text{.} \end{equation*}

(Make sure you replace the question mark by a positive exponent.)

Hint.

Task 2.11.a.

(c)

In Task b you have established that there is some positive exponent \(n\) so that \(g^n = e\text{.}\) Now convince yourself that in this case there is always a smallest such positive exponent.