MTH3003 Lecture 5

Following on from permutations and subgroups, we can look at some groups that arise from geometry. Specifically, we can look at the symmetries of regular $n$-gons, which form an important class of groups known as dihedral groups.

Symmetries of a Square

We will start by looking at a regular 4-gon (a square) drawn in the Euclidean plane with its centre at the origin. By looking at the operations we can perform without changing the square, we find only 8 symmetries:

• 4 reflections (through 4 lines of reflectional symmetry)
• 4 anticlockwise rotations (by angles $0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$)

Because these operations send corners to corners, we can label the corners $\{1,2,3,4\}$ and view these symmetries as permuting the corners. This set of symmetries, under the operation of combining symmetries, forms a group called the symmetry group of the square, or the dihedral group of order 8. Because it permutes four corners, we can think of it as a subgroup of $S_4$.

Dihedral Groups

We can generalise this to any regular $n$-gon for $n\ge 3$. Labeling the corners anticlockwise $1,2,\dots ,n$, we can make four key observations:

1. The $n$-gon has at most $2n$ symmetries.
2. It has at least $n$ lines of reflectional symmetry.
3. It has at least $n$ rotational symmetries (permuting corners as $$\begin{gathered} (1 \\ ,\dots n) \end{gathered}$$).
4. No rotational symmetry is a reflectional symmetry, and vice versa.

The Dihedral Group $D_{2n}$

The symmetry group of a regular $n$-gon with corners labelled $1,2,\dots ,n$ has $2n$ elements and is written $D_{2n}=\{e,\rho ,{\rho }^2,\dots ,{\rho }^{n-1},{\sigma }_1,\dots ,{\sigma }_n\}$, where $$\begin{gathered} \rho =(1 \\ ,\dots n) \end{gathered}$$ and the ${\sigma }_i$ are reflections. This is the dihedral group of order $2n$, which is a subgroup of $S_n$.

We can prove this is a group by applying the Quick Subgroup Test: the identity $e\in D_{2n}$, it is closed (applying two symmetries gives a symmetry), and every element has an inverse (reflections are their own inverse, and ${\rho }^{-i}={\rho }^{n-i}$).

Note: Some mathematicians and textbooks use $D_n$ instead of $D_{2n}$ to refer to this same group. Most use $D_{2n}$, but it’s important to be careful.

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Pre-Lecture Notes from University Notes

• Groups arising from geometry - looking at the symmetries of regular $n$-gons.
• The dihedral group of order 8 (symmetries of a square), mapped to permutations in $S_4$.
• Generalised dihedral group $D_{2n}$ of order $2n$, defined by $n$ rotations and $n$ reflections.
• Proof that the number of symmetries of a regular $n$-gon is $2n$.