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Determining isomorphic summetry groups

Posted on 2011-03-17
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I have three shapes, the group for each shape is a symmetry group. I need to determine which two groups are isomorphic.

The shapes are a triangle, a square, and a four pointed star (which is essentially a square with a triangle stuck on each side).

Visually the square and the four pointed star appear to be isomorphic, but how to I show this?
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Question by:purplesoup
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by:TommySzalapski
ID: 35158523
What do you mean by "the group for each shape"? Isomorphism just means "structually the same" so it kind of depends on definition. Usually, in math, things are considered isomorphic if you can "morph" one into the other without making any significant changes. With geometric shapes, I would assume that isomorphic was used to mean that they were the same regardless of scaling or rotation. So if the group is all the same except some are bigger or rotated, I would say they are isomorphic.
If the proportions or angles are different they are not.
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by:purplesoup
ID: 35158959
I mean symmetrical operations performed on the shape - so the points of the shape are numbered, and then transformed through various axis of symmetry.

The star and the square have the same axis of symmetry so I think they are isomorphic, but I don't know what would constitute an acceptable proof for this.
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by:TommySzalapski
ID: 35159002
Just list the symmetrical operations required.
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by:ozo
ID: 35160541
The square and the star are both D4
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by:purplesoup
ID: 35165999
I don't know what D4 is.

Is listing symmetrical operations a proof of isomorphism? The examples I have seen of showing two groups are isomorphic is to map each element from one group to the other group.

If listing symmetrical operations (e.g. reflections) is a good proof, the way I have seen symmetrical operations denoted is to mark them (e.g. reflections) on a diagram and label them with letters.

If this was done on both shapes, would mapping the letters of different operations be an acceptable way of showing isomorphism?
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TommySzalapski earned 500 total points
ID: 35167586
Yes. That is how I would do it. That might not be considered a rigorous proof, but I'm positive you don't need one.
For one thing you've been using words like 'determine' and 'show' which imply that you don't need a super formal proof. Also, if the shapes aren't formally and rigorously defined then you can't anyway. The method you propose is the best (in my opinion) for this particular question.
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by:TommySzalapski
ID: 35167617
From http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Nazarewicz/7210_final_2/7210_Project/index.html
An object with D4 symmetry would have four rotations, each of 90 degrees, and four reflection mirrors, with each angle between them being 45 degrees.

So since both objects have the same symmetry, they should be isomorphic. But this is an observation, not a proof, and if you haven't been given that as a tool, then you probably shouldn't use it (assuming this is an academic excercise).
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