# Maths - Dihedral Groups - Datasheet

## Cayley Table

The Cayley table has four quadrants shown here in red,green,blue and orange, each of these quadrants is similar to the Cayley table for a cyclic group:

 {1,1} {r,1} {r²,1} {1,m} {r,m} {r²,m} {r,1} {r²,1} {1,1} {r,m} {r²,m} {1,m} {r²,1} {1,1} {r,1} {r²,m} {1,m} {r,m} {1,m} {r,m} {r²,m} {1,1} {r,1} {r²,1} {r,m} {r²,m} {1,m} {r,1} {r²,1} {1,1} {r²,m} {1,m} {r,m} {r²,1} {1,1} {r,1}

## Cayley Graph

There are two cycles that go in opposite directions

## Cyclic Notation

The two generators are:

1. A complete cycle
2. A set of 2-element cycles which swap opposite points

(1,2,3..n) , ( 1 n ) ( 2 (n-1)… )

## Group Presentation

Thereare two generators, r (for rotate) is like a cyclic group, and m (for mirror) which flips the shape over.

<r , m | rn =1,r² = 1, m r m= r-1>

or equivalently:

<x , y | x² = y² = xyn = 1>

## Group Representation

One generator is a rotation of the identity matrix through 90° (the diagonal from bottom-left to top-right), the other is the nth root of the identity matix (such that lesser roots are not identity). See this page for information about taking roots of a matrix. One a matrix which will do this is an n×n matrix on the right:

 0 0 … 0 0 1 1 0 … 0 0 0 0 1 … 0 0 0 0 0 … 1 0 0 0 0 … 0 1 0
,
 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0

## Related datasheets

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 Symmetry and the Monster - This is a popular science type book which traces the history leading up to the discovery of the largest symmetry groups.

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