Proving Dn Non-Abelian for n >= 3

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Therefore, any element of Dn can be written as a composition of rotations and reflections, and since rotations and reflections don't commute, Dn is not abelian. In summary, the goal is to prove that if n >= 3 then Dn is non-abelian by showing that some symmetries in Dn are not commutative and finding an isomorphism to Dn that is not abelian. The approach is to use the fact that rotations and reflections do not commute to show that Dn is not abelian.
  • #1
zcdfhn
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My goal is to prove that if n >= 3 then Dn, which is the dihedral group of size n is non-abelian.

I am stuck, but my attempt at a solution was to show that some the symmetries in Dn are not commutative under composition, but I do not know how to prove that for n cases. My other attempt was to find a isomorphism to Dn and show that its isomorphism is not abelian, but I have no clue where to start from there.

I would appreciate either an answer, or something to help me get going. Please be precise and I thank you so much in advance.
 
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  • #2
In general, a rotation doesn't commute with a reflection.
 

Related to Proving Dn Non-Abelian for n >= 3

1. What does it mean for Dn to be non-abelian?

For a group to be non-abelian, it means that the order in which the group elements are multiplied together matters. In other words, the group is not commutative, meaning that a*b is not always equal to b*a.

2. Why is it important to prove that Dn is non-abelian?

Proving that Dn is non-abelian is important because it helps us understand the structure and properties of this group. It also allows us to distinguish Dn from other groups, and use this property in mathematical applications.

3. How can we prove that Dn is non-abelian for n >= 3?

There are a few different methods for proving that Dn is non-abelian for n >= 3. One way is to show that there exists a specific example where the order of multiplication matters, thus demonstrating non-commutativity. Another method is to use the group presentation of Dn and show that the defining relations do not satisfy the commutative property.

4. Can Dn be non-abelian for n < 3?

No, Dn is only non-abelian for values of n greater than or equal to 3. For n = 1, Dn is isomorphic to the trivial group, which is abelian. For n = 2, Dn is isomorphic to the cyclic group of order 2, which is also abelian.

5. What implications does the non-abelian property of Dn have in other mathematical contexts?

The non-abelian property of Dn has various implications in different areas of mathematics. For example, in group theory, it allows us to classify and differentiate between different types of groups. In geometry, it helps us understand the symmetries of regular polygons. In physics, non-abelian groups are used to describe the fundamental interactions between particles in the Standard Model.

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