Quick definition question: Dihedral group

Click For Summary

Discussion Overview

The discussion revolves around the definition and properties of dihedral groups, specifically focusing on the Dihedral group of order 2n, denoted as Dn. Participants explore the symmetries of polygons, particularly squares and other n-gons, and discuss methods for constructing multiplication tables for these groups.

Discussion Character

  • Technical explanation, Mathematical reasoning, Conceptual clarification

Main Points Raised

  • One participant states that a dihedral group of an n-gon, denoted by Dn, has an order of 2n, providing examples of symmetries for a square (8), octagon (16), and hexagon (12).
  • Another participant confirms the initial claim regarding the order of the dihedral groups.
  • A participant inquires about the necessity of labeling reflections when constructing the multiplication table for D4 and seeks an easier method for this task.
  • A later reply suggests using specific relations (S^2 = 1, R^4 = 1, SR = R^3S) to facilitate the construction of the multiplication table for D4.

Areas of Agreement / Disagreement

Participants generally agree on the basic properties of dihedral groups and their orders, but there is no consensus on the easiest method for constructing multiplication tables, as different approaches are discussed.

Contextual Notes

Some assumptions about the understanding of group theory and notation may be implicit in the discussion, and the methods for constructing multiplication tables may depend on individual familiarity with the concepts.

srfriggen
Messages
304
Reaction score
7
A dihedral group of an n-gon denoted by Dn, whose corresponding group is called the Dihedral group of order 2n?


What I gather from that is a square has 8 symmetries, an octagon has 16, a hexagon 12, etc?
 
Physics news on Phys.org
Yep, you got it.
 
When I do reflection in square do I need to put numbers and that do reflection for particular site of square. Is there some easiest way to find multiplication table of ##D_4##?
 
LagrangeEuler said:
When I do reflection in square do I need to put numbers and that do reflection for particular site of square. Is there some easiest way to find multiplication table of ##D_4##?

Let ##R## be a rotation and ##S## a reflection. Use the relations ##S^2 = 1##, ##R^4 = 1## and ##SR = R^3S##. Using this you can find the multiplication table easily.
 

Similar threads

Undergrad Dfns group action & group, written in terms of the other
  • · Replies 26 ·
Replies
26
Views
1K
Graduate What Are the Properties of Dihedral Groups?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
982
Graduate What are the properties of a dicyclic group?
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Why can't Dn be isomorphic to the direct product of its subgroups?
  • · Replies 2 ·
Replies
2
Views
6K
MHB What Is the Upper Bound of Groups of Order in Finite Group Theory?
  • · Replies 2 ·
Replies
2
Views
2K
Showing that a subgroup of Sym(4) is isomorphic to D_8
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Structure of modulo-integer matrix group GL(r,Z(n))?
  • · Replies 17 ·
Replies
17
Views
7K
Graduate Tensor symmetries and the symmetric groups
  • · Replies 1 ·
Replies
1
Views
2K
Proving that the Dihedral Groups are non-cylic
  • · Replies 5 ·
Replies
5
Views
2K