Group theory

Date: Tue, 22 Nov 1994 08:21:15 -0800
From: (Matthew Baker)
Subject: Question

Could you please answer this??

The four rotational symmetries of the square satisfy the four requirements 
for a group, and so they are called a subgroup of the full symmetry group.  
(Notice that the identity is one of these rotational symmetries and that 
the product of two rotations is another rotation in the subgroup.)

   a. Do the four line symmetries of the square form a subgroup?

   b. Does the symmetry group of the equilateral triangle have a subgroup?

Date: Mon, 28 Nov 1994 11:22:49 -0500 (EST)
From: Dr. Sydney
Subject: Re: Question

Hello there!  Thanks for writing Dr. Math!  What a good question you are
asking...

When you ask about the four line symmetries, I assume you are talking about
reflections about the 4 axes of symmetry, right?  Well, it turns out these 4
reflections will not be a subgroup of the total symmetry group because the
set of all reflections is not closed.  Can you see why?  If you draw a
picture and label axes and vertices, and try and perform two of these
reflections, you'll find you will not get another reflection, but instead
you'll get a rotation.  Try it out!  Another reason they don't form a
subgroup is because the identity element is not an element of the set.
However, if you take the set that contains the identity element and any ONE
reflection, you will get a subgroup.  Can you see why?

On to your next question...the symmetry group of the equilateral triangle
does have subgroups; in fact it has 4 subgroups in addition to the trivial
subgroups.  Can you guess what they may be?  (Hint: very similar to the
symmetry group of the square--consider rotations and reflections.) See if
you can figure this one out--if you want you can write back with your
thoughts on the problem--I'd love to hear what you think.

I'm quite impressed you are doing group theory in high school--I am just
learning group theory this semester in an abstract algebra class.  Please do
write back with any more questions you might have. 

--Sydney, a math doctor