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Two questions related to Dihedral groups:

  1. What is the conventional notation for Dihedral groups? Is it Dn where n is the number of sides in a regular n-gon, or is it D2n where n is the number of symmetries in the n-gon? AQA haven't specified for the Further Maths A-Level which is concerning me as you could get asked about D6 without the question specifying what it is, a hexagon or an equilateral triangle dependent on different definitions. What is conventional and what would you expect them to define Dn as?

Next is a question:

Show that the set of symmetries of an equilateral triangle forms a group:

Let the point of intersection of the lines of symmetry of an equilateral triangle be the origin

Let r0 = rotation of 0 degrees about the origin (i.e. the triangle is in its initial orientation).

Let r1 = rotation of 120 degrees anticlockwise about the origin

Let r2 = rotation of 240 degrees anticlockwise about the origin.

For triangle equilateral triangle ABC (C bottom left point, B bottom right):

m1 = reflection in the mirror line through A m2 = reflection in the mirror line through B m3 = reflection in the mirror line through C:

The question then expects a Cayley table to be drawn showing the combination of the different symmetries:

NOTE: Unsure of how to create a Cayley table using LaTeX, or a table of contents so these are the values of the table I have been able to work out:

r0 is the identity element

r1 followed by r1 is r2

r1 followed by r2 is r0 and vice versa

r2 followed by r2 is r1

mn followed by mn is r0 for n = 1, 2, 3.

I am unsure of other values though, for example, I can't understand why r1 followed by m1 is m2.

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    $\begingroup$ Write $(123)$ for the rotation sending vertex $1$ to $2$ and $2$ to $3$ and $3$ to $1$, respectively $(12)$ for the reflection sending $1$ to $2$ and $2$ to $1$, This generates the group $D_3=S_3$. Now just use the Cayley table of $S_3$. $\endgroup$ – Dietrich Burde Feb 28 at 20:29
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It depends on what book you are using. I always think of $D_n$ as the group of symmetries of a regular $n$-gon, a group which has $2n$ elements. But some also write this group as $D_{2n}$. So here you have to decide for yourself what you prefer.

As for the second question, just draw a picture. Draw a triangle with vertices $a,b,c$ and look where does each vertex move when you do $r_1$ followed by $m_1$.

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  • $\begingroup$ Thank you, and I've tried to do that but, in the case of r1 followed by m1, when you. Under r1, triangle ABC -> triangle BCA and under m1 -> triangle BAC, which isn't the same as a reflection in m2. $\endgroup$ – Sam Connell Feb 28 at 20:31
  • $\begingroup$ You are right, it equals the reflection $m_3$, not $m_2$. Why did you think it should be equal to $m_2$? $\endgroup$ – Mark Feb 28 at 20:44
  • $\begingroup$ The textbook explanation for the question. I'm using the column as the leftmost column of the Cayley table as the first reflection/rotation though. If m1 followed by r1 is the same as reflection in m2 then it's a simple misunderstanding, although I always thought it was convention to use the leftmost column as the first value and the top row as the second in a Cayley table. I think my teacher has said before it doesn't matter as long as you're consistent though. $\endgroup$ – Sam Connell Feb 28 at 21:22
  • $\begingroup$ Yes, $m_1$ followed by $r_1$ is indeed $m_2$. The order of operations matters. There are different ways to write a Cayley table, that's true. $\endgroup$ – Mark Feb 28 at 21:33

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