Two questions related to Dihedral groups:
- 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.