Maths - Dihedral Groups - Datasheet

For more information about dihedral groups see this page.

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}

For more information about Cayley table see this page.

Cayley Graph

There are two cycles that go in opposite directions

For more information about Cayley graph see this page.

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)… )

For more information about cyclic notation see this page.

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>

For more information about group presentation see this page.

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

For more information about group representation see this page.

Related datasheets


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see also:

 

Correspondence about this page

Book Shop - Further reading.

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.

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Terminology and Notation

Specific to this page here:

 

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