# One more group Question

• Dec 15th 2007, 09:04 PM
Amy
One more group Question
Yesterday I posted a qn. on subgroups. The answers really helped me.
I have one more ..

I didn't understand this qn. at all

What does it mean and how to do it?
• Dec 15th 2007, 10:33 PM
topsquark
Quote:

Originally Posted by Amy
Yesterday I posted a qn. on subgroups. The answers really helped me.
I have one more ..

I didn't understand this qn. at all

What does it mean and how to do it?

Sketch an equilateral triangle and plot a point at the center of it.

Note that if you rotate the triangle 120 degrees about this point the triangle looks the same as it initially did. This is the $\displaystyle R_{120}$ symmetry. Similarly for $\displaystyle R_{240}$. ($\displaystyle R_0$ would be the "identity" transformation. Obviously if you rotate the triangle by 0 degrees it looks the same as it did before you "rotated" it.)

For the "flips" sketch a line from the center of the triangle through vertex A. Note that we have reflection symmetry over this line: if we reflect the triangle over this line it looks the same as it did before. This is the element $\displaystyle F_A$. Obviously we have the same kind of symmetry no matter which vertex we pick, so we also have $\displaystyle F_B$ and $\displaystyle F_C$. (The $\displaystyle F_H$ and $\displaystyle F_V$ are symmetries about "horizontal" and "vertical" lines. These symmetries depend on the orientation of the square, obviously.)

See what you can do with these rotations and reflections for a pentagon. (Hint: You'll have 5 rotations and 5 reflections.)

-Dan
• Dec 16th 2007, 10:05 AM
Amy
Thanks for the reply.
So, is it some thing like
s={R0, R72,R154,R226,R298, FA, FB, FC,FD,FE}??
• Dec 16th 2007, 10:28 AM
topsquark
Quote:

Originally Posted by Amy
Thanks for the reply.
So, is it some thing like
s={R0, R72,R154,R226,R298, FA, FB, FC,FD,FE}??

That'll do it!

Another one to watch out for (not present here) is "inversion" symmetry. The best way to describe this one is to let the center point of your figure be the origin. A group has inversion symmetry if you perform $\displaystyle x \to -x$ and $\displaystyle y \to -y$ and the figure looks the same as it did before the operation. The typical symbol for inversion is "i."

-Dan
• Dec 18th 2007, 04:16 AM
JaneBennet
In general, the symmetry group of a regular n-gon will consist of n rotations and n flips. The rotations will be through a mulitple of $\displaystyle \frac{360}{n}$ degrees. For the flips, it depends on whether n is odd or even. If n is odd, all the flips will be about axes passing from one vertex to the midpoint of the opposite side. If n is even, half the flips will be about axes passing from one vertex to the opposite vertex and the other half will be about axes passing from the midpoint of one side to the midpoint of the opposite side.

The symmetry group of the regular n-gon (n ≥ 3) is called the dihedral group of order 2n. Also, flips are more usually called “reflections”. :D

So, for the regular pentagon:

Quote:

Originally Posted by Amy
Thanks for the reply.
So, is it some thing like
s={R0, R72,R154,R226,R298, FA, FB, FC,FD,FE}??

That’s correct apart from a minor error with the rotation angles. Check your calculations again. :)