Group Theory: Rotational Symmetries

In summary, group theory is a branch of mathematics that studies symmetries and structures of objects, particularly rotational symmetries. It is used in science to describe the symmetries of physical systems and plays a key role in understanding quantum mechanics. In group theory, a group action is a way of assigning elements to objects while preserving structure. Real-world examples of rotational symmetries can be found in natural and man-made objects such as circles, snowflakes, and wheels, as well as in the design of buildings and furniture.
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Homework Statement



Show that the group R of rotational symmetries of a dodecahedron is simple and has order 60.


The Attempt at a Solution



I see how to get order 60 using the orbit stabilizer theorem. Letting R act in the natural way on the set of faces, we find the size of the orbit of a face is 12 and the size of the stabilizer of a face is 5. So |R| = 60.

Also, by letting R act on the set of cubes inscribed inside the dodecahedron, we can show that R is isomorphic to a subgroup of S5, so must be A5, which is the only order 60 subgroup of S5. But I don't think I'm supposed to use this method because the next question is to show that R is isomorphic to A5.

Anyway, how do you determine that R is simple without having to totally classify it?

Thanks in advance.
 
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Thank you for your question. To show that the group R of rotational symmetries of a dodecahedron is simple, we can use the same approach as you did in finding the order of the group. By letting R act on the set of faces, we can see that the orbit of a face is 12 and the stabilizer of a face is 5. This means that R is a subgroup of S12, the symmetric group on 12 elements. Since |R| = 60, R must be isomorphic to a subgroup of S5, which is isomorphic to A5.

To show that R is simple, we need to prove that it has no non-trivial normal subgroups. One way to do this is by using the fact that A5 is simple. Since R is isomorphic to A5, any normal subgroup of R must also be a normal subgroup of A5. But since A5 is simple, it has no non-trivial normal subgroups, which means that R also has no non-trivial normal subgroups. Therefore, R is simple.

I hope this helps. Let me know if you have any further questions.


 

FAQ: Group Theory: Rotational Symmetries

1. What is group theory?

Group theory is a branch of mathematics that studies the symmetries and structures of objects. It deals with the ways in which objects can be transformed while preserving their essential properties.

2. What are rotational symmetries?

Rotational symmetries are transformations that preserve the shape and size of an object while rotating it around a fixed point. In group theory, these symmetries are represented by mathematical objects called groups, which consist of elements and operations.

3. How is group theory used in science?

Group theory has many applications in science, particularly in physics and chemistry. It is used to describe the symmetries of molecules, crystals, and other physical systems. It also plays a key role in understanding the behavior of particles and waves in quantum mechanics.

4. What is a group action?

In group theory, a group action is a way of assigning elements of a group to objects or structures in a way that preserves the group's structure. This can be thought of as a transformation that is performed on a set of objects.

5. What are some real-world examples of rotational symmetries?

Rotational symmetries can be found in many natural and man-made objects. Some common examples include the symmetry of a circle, the symmetry of a snowflake, and the symmetry of a wheel. They can also be seen in the design of buildings, furniture, and other objects that have rotational symmetry.

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