Dihedral group element composition definitions

Stream: maths

[Ben]

Getting a little confused by the dihedral group definition and what r and sr are... these *facts* here don't seem to line up with the ones on wikipedia (yah not a great source ik). Are the ones on wikipedia incorrect or referring to different things... image.png. In mathlib dihedral_group.r i * dihedral_group.sr j = dihedral_group.sr (j - i) whereas wikipedia states r^i * s^j=s^{i+j}?

[Adam Topaz]

These formulas depend on how you define s_j. If we take the "standard" presentation to be
$D_{2n} = \langle r,s \ | \ r^n = s^2 = (rs)^2 = 1 \rangle$
and define $r_j = r^j$ and $s_j = r^j \cdot s$, then we would have $r_i \cdot s_j = s_{i+j}$.

But if we define $s_j = s \cdot r^j$ then we would have $r_i \cdot s_j = r_i \cdot s \cdot r^j = s \cdot r^{j-i} = s_{j-i}$.

[Adam Topaz]

It's all just a question of what you mean by the "j-th" reflection and the "i-th" rotation. You can parameterize these things in different ways.

[Ben]

hmm never seen that chevron construction for the dihedrals before...
So does the mathlib def sr i mean a reflection then a rotation of i? Both wikipedia and mathlib say that s or sr is a "reflection".

Also what is the difference between $r_i$ and $r^i$...?

[Adam Topaz]

What is your definition of D_{2n}?

[Ben]

the dihedral group... wait you can define the $D_{2n}$ in different ways...

[Kevin Buzzard]

There are loads of ways to define it, which is why your question is so hard to understand

[Kevin Buzzard]

Different definitions, especially using similar or the same notation to mean different things, can lead to confusion if you don't understand the material properly

[Ben]

I think I like the lean definition, where you can assume an element is either one of two variants.

[Adam Topaz]

Okay, so you want to define $D_{2n}$ as some type with $2 \cdot n$ elements, denoted
$\{r_0,\ldots,r_{n-1},s_0,\ldots,s_{n-1}\}$,
but now the question is how you want to define the product!

[Junyan Xu]

Ben said:

So does the mathlib def sr i mean a reflection then a rotation of i?

By usual convention, where multiplication means function composition, I think it means a rotation then a reflection: when you apply a*b, b is first applied, then a.

Both wikipedia and mathlib say that s or sr is a "reflection".

Yes, in a dihedral group, the product of a reflection with a rotation (to the left or to the right) is always a reflection.

Also what is the difference between $r_i$ and $r^i$...?

I think there's no difference.

[Ben]

Yah I guess it's like how Peano numbers are only one way of defining the natural numbers... didn't realise there were different forms, especially when they use the the same words and similar terms...

[Ben]

So the element composition definitions on wikipedia are correct, they are just using a different definition (which they don't define, just call it a reflection)

[Ben]

What does the s and r in the sr constructor mean in the definition on mathlib... how does it match up geometrically on a ngon?

[Kevin Buzzard]

That depends on how you're defining the group law on the n-gon. If you do reflection $a$ and then reflection $b$, have you just done $ab$ or $ba$? Both answers are "valid" in the sense that both will give you groups, but the groups won't be identical (they will be isomorphic, but the isomorphism won't be the identity function)

[Adam Topaz]

If you have the regular $n$-gon, centered in the plane, with a vertex on the $y$-axis, you can define $r$ to be rotation counter-clockwise by $2 \cdot \pi /n$ radians (equiv. multiplication by $\exp(2 \cdot \pi \cdot i / n)$ in the complex plane. Also define $s$ to be reflection about the $y$-axis.

With these conventions, the collection of all symmetries of the regular $n$-gon are $\{r^0,r^1,\ldots,r^{n-1},r^0s,r^1s,\ldots,r^{n-1}s\}$. The multiplication rules follow from the formulas: $r^n = s^2 = (rs)^2 = 1$ (left as an exercise).

[Adam Topaz]

In this case, you can define $r_i = r^i$ and $s_i = r^i \cdot s$ to get $r_i \cdot s_j = s_{i+j}$ (arithmetic being done modulo $n$).

[Kevin Buzzard]

Adam the convention in mathlib is that sr i means $sr^i$ so doing $r^js$ is only adding to the confusion.

[Adam Topaz]

Alternatively, note that $r^i s = s r^{-i}$ so you can equivalently view $D_{2n}$ as the set $\{r^0,r^1,\ldots,r^{n-1},sr^0,sr^1,\ldots,sr^{n-1}\}$ and in this case if you define $s_j = s r^j$ then you have $r_i s_j = s_{j-i}$.

[Adam Topaz]

@Kevin Buzzard yeah that's kind of the point, since wikipedia seems to use $s_j = r^j s$.

[Kevin Buzzard]

Oh I see, you were half way through an explanation :-) Yeah, both of these work, and interpreting them as acting on the n-gon you have to decide whether you're acting on the left or the right (or equivalently whether $ab$ means "do $a$ then $b$" or "do $b$ then $a$") which introduces yet another sign ambiguity.

[Adam Topaz]

as Junyan's message mentioned, the "standard" convention is that $a \cdot b$ first acts by $b$ then by $a$.

[Adam Topaz]

So for example $sr^2$ will first apply the rotation $r$ twice, then apply the reflection, whereas $r^2s$ will first apply $s$ then apply $r$ twice.

[Junyan Xu]

So if you map r i in mathlib to $r_i$ in Wikipedia and map sr i to $s_{n-i}$ ("$s_{-i}$"), you get an isomorphism between the two dihedral groups.