Simplifying f=uRuR in D3: A Beginner's Guide

[morrowcosom]

Homework Statement

In D3, simplify f = u R u R

Homework Equations

The Attempt at a Solution

u(Ru)R=(uR)uR
u(uR^2)R=(uR)u2
uR=R

I am doing this for independent study, and the website I am using says the solution is 1=R

I just got started on dihedral groups and have no idea what in the heck to do. How do you tell which is a rotation or reflection before the solution arises and/or arrive with the above answer of R=1

Thanks

 

[HallsofIvy]

I have no idea what the notation "u R u R" means. Are u and R members of the group or something else?

 

[morrowcosom]

I have no idea what the notation "u R u R" means. Are u and R members of the group or something else?

That is the way the problem is stated. Here are some relevant equations and an example problem.

Relevant equations:
To describe the group operation of Dn, we need only note the following rules (called "relations"):
u^2 = 1 (in fact, uk^2 = 1 for every reflection uk)
Rn = 1 (so R-1 = R^n-1)
R u = u R-1 = u R^n-1

To make calculations unambiguous, we agree that every element of Dn will have a standard form:

Rotations: Rk (or 1 for the identity = R0)
Reflections: u Rk (or just u itself if k = 0)
Since Rn = 1, we can take k between 1 and n-1

Example problem:
Example 1: In D3, the symmetry group of the equilateral triangle, calculate a = u R2 u.

Here is a sequence of steps using the relations in D3 along with associativity:
a = u R2 u = (u R2) u = u (R2 u)
= u (R R u) = u R (u R2)
(note we are using relation 3 above, for n = 3)
= u (R u) R2 = u u R4
(now use relations 1 and 2)
= u 2 R4 = R
So we conclude: a = u R2 u = R in D3