Group theoryDate: 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 |
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