Short description about dihedrals?

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Discussion Overview

The discussion revolves around the concept of dihedral groups, specifically seeking a short description and exploring their elements, such as those in D10 and D4. Participants examine both geometric interpretations and abstract algebraic representations of these groups.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant requests a brief description of dihedrals and their elements, specifically D10 and D4.
  • Another participant explains that dihedral groups can be interpreted geometrically as consisting of rotations and reflections.
  • A question is raised about whether the geometric interpretation applies similarly in abstract algebra.
  • It is affirmed that the geometric interpretation serves as motivation for dihedral groups, with a group presentation provided for D_n.
  • A participant notes that the group presentation includes the relation sr = rs^{-1}.
  • Another participant reiterates the need for the relation sr = rs^{-1} and acknowledges a mistake regarding the finiteness of the group for n>1.
  • It is mentioned that the group is actually the free product of C_n and C_2.

Areas of Agreement / Disagreement

Participants generally agree on the geometric interpretation of dihedral groups and the elements involved, but there are nuances regarding the algebraic representation and the nature of the group for n>1 that remain unresolved.

Contextual Notes

There are limitations regarding the assumptions made about the finiteness of the group and the specific definitions of the elements in the context of dihedral groups.

vivaitalia1
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Can someone give me a short description about dihedrals? for example what would be the elements of D10? or D4?
 
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Well, do you know the geometric interpretation of a dihedral group? The elements are (in the geometric interpretation) rotations and reflections.
 
Yes I know the geometric interpretation. Is it the same for abstract algebra?
 
Yes, the geometric interpretation is the motivation for dihedral groups. As I said, the elements are analogous to rotations and reflections. In general a group presentation for D_n is

\langle s, r| s^n = 1, r^2 = 1 \rangle.

In that presentation, s corresponds to a rotation and r to a particular reflection.
 
Last edited:
You also need sr = rs^{-1} in that presentation.
 
morphism said:
You also need sr = rs^{-1} in that presentation.

Indeed! I should be more careful. The group in my post isn't even finite for n>1.
 
Last edited:
It is the free product of C_n and C_2, in fact.
 

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