Proving Non-Cyclic and Non-Abelian Properties of Dihedral and Symmetric Groups?

[MikeDietrich]

Homework Statement

Prove that S_n and D_n for n>=3 are non-cyclic and non-abelian.

Homework Equations

I get that I need to show that two elements from each group do not commute and that there is not a single generator to produce the groups... I am just unsure of how to do this.

The Attempt at a Solution

I can prove the non-abelian part for both by "multiplying" two of the elements and showing they do not commute (easy enough). However, how do I generalize the property to all n>=3? I have no idea where to start with the cyclic part. I have a hard time visualizing how rotations and reflections could be cyclic.

Any guidance appreciated. I generally find someone says one little thing that makes everything click into place. Thanks!

 

[micromass]

Well, it suffices to show that the groups are not abelian. This implies that the groups are not cyclic (since any cyclic group is abelian).

Let's start of with Sn. You need to find two elements in Sn which do not commute. Can you find me such elements in S3??

 

[MikeDietrich]

Sure. r = rotation, s = refection
Let's say r_1=3, r_2=1, r_3 = 2 and s_1 = 2, s_2 = 1 and s_3 = 3 then (r)(s) = (1,3,2) and (s)(r) = (3,2,1). Obviously, they do not commute. How do I generalize that to include all n>=3?

 

[micromass]

Well, I think this example generalizes to all n... It will never be true that rs=sr.

 

[MikeDietrich]

Yeah... you're right. I was over-thinking this one. THANKS!

 

[Outlined]

Is is enough to show D_n is non-abelian as it is a subgroup of S_n. This follows directly from a definition (note s^-1 = s)

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