UA-AUG-AUMAT229 Dihedral groups: Discovery guide

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.

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 $D_{3} .$

Here are three of the six elements of $D_{3} :$

the trivial rotation of $0$ radians;
rotation about the $z$ -axis by $2 π / 3$ radians counter-clockwise (when looking “down” the $z$ -axis from “above”);
rotation about the $x$ -axis by $π$ 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 $s .$

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(a)

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

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(b)

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

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(c)

Fill in the multiplication table for $D_{3} .$

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 $e,$ $r,$ or $s,$ 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.

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(d)

Does order of multiplication matter in $D_{3} ?$ In particular, use your table to compare the results of $r s$ and $s r$ — are they the same?

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(e)

Use your multiplication table to determine $r^{- 1}$ and $s^{- 1} .$ Then use algebra and the pattern of Task 2.10.b to determine the inverses of each of the other elements of $D_{3},$ and check your table to make sure your results are correct.

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(f)

In carrying out algebraic calculations with the elements in $D_{3},$ 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

s^{2} = e .

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

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(g)

Are there any other non-algebraic patterns you notice in the multiplication table for $D_{3} ?$

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For an element

g

in a (multiplicative) group

G,

we call the smallest positive integer

n

for which

g^{n} = e

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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

n \cdot g = 0

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(again, if such an

n

exists).

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Discovery 4.2.

Use your multiplication table to determine the order of each of the elements in $D_{3} .$

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Discovery 4.3.

Consider 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 $D_{4} .$

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(a)

Based on your experience with $D_{3},$ which rotation of this square plate should be denoted $r,$ and which should be denoted $s ?$

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(b)

Carry out an analysis of $D_{4}$ similar to that carried out for $D_{3}$ in Discovery 4.1 and Discovery 4.2.

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Discovery 4.4.

In general, a flat, thin, solid whose face is a regular $n$ -agon has a dihedral rotational symmetry group, $D_{n} .$ Consider the algebraic patterns that $D_{3}$ and $D_{4}$ had in common, and conjecture how these patterns might generalize to $D_{n} .$

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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)?

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Discovery 4.6.

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

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(a)

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

e, g, g^{2}, g^{3}, \dots

cannot contain an infinite number of different results.

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(b)

Suppose $k$ and $m$ are positive integers so that

g^{k} = g^{m}

but $k \neq m .$ (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

g^{?} = e .

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

Hint.

Task 2.11.a.

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(c)

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