# D_6

• Sep 13th 2011, 05:44 PM
dwsmith
D_6
How do I find the order of the elements dihedral group \$\displaystyle D_6\mbox{?}\$. My book doesn't explain how to find the order of the elements.

Thanks.
• Sep 13th 2011, 05:55 PM
TheChaz
Re: D_6
Well, take an element of D_6, and compose it with itself until you get the identity.

Each of the rotations has order 3 (if 6 = 2n)
Each of the reflections has order 2.
• Sep 13th 2011, 06:48 PM
dwsmith
Re: D_6
Quote:

Originally Posted by TheChaz
Well, take an element of D_6, and compose it with itself until you get the identity.

Each of the rotations has order 3 (if 6 = 2n)
Each of the reflections has order 2.

Can you be more specific or give an example?
• Sep 13th 2011, 06:57 PM
TheChaz
Re: D_6
Quote:

Originally Posted by dwsmith
Can you be more specific or give an example?

Let's change assumptions here and suppose that D_6 is the dihedral group of the regular hexagon (not triangle).
Dihedral Group D6 -- from Wolfram MathWorld

In the article, x is a rotation of 60 degrees. So....
x*x is a rotation of 120 degrees.
x*x*x is 180 degrees
.
.
.
x^6 is 360 degrees = 0 degrees, so x^6 is the identity. Being the smallest such exponent (for x), SIX is the order of x.
If you repflect (the operation "y" in the article), then reflect again, you get back to where you started.

What is your understanding of the order of an element in a Group (in general) ? Methinks there is something lacking.
• Oct 2nd 2011, 04:43 PM
dwsmith
Re: D_6
The elements of \$\displaystyle D_6=\{1,r,r^2,s,rs,r^2s\}\$

Orders:
\$\displaystyle |1|=1\$
\$\displaystyle |r|=3\$
\$\displaystyle |r^2|=\mbox{?}\$ Order 2?
\$\displaystyle |s|=2\$
\$\displaystyle |rs|=\mbox{?}\$ Order 2?
\$\displaystyle |r^2s|=\mbox{?}\$ Order 2?

How do I find the other orders? I am trying to understand this but my book says nothing about it.