Maths - Generators - Combinatorial Group Theory

This page continues our discussion of finite and permutation groups from this page. Here we look at how to define and categorise the groups in a more formal way.

Ways to define the structure of a group

What is the minimum information required to define the structure of this group? We could:

But is there any other way?

Presentation of a group

It turns out that we can specify the group by its generators plus a set of relations among the generators (relations that produce the identity element are usually written without the equals sign).

This is known as the 'Presentation of a group'.We specify this as follows:

Γ< generators | relations >

Using the triangle example, from this page, we have:

Γ< a,b | aa,bb,bab=aba>

We could have set all relations to be equal to unity by replacing bab=aba with baba-1b-1a-1=1

Can we use this to generate all the entries in the table and thus completely define the structure?

Lets take an example of multiplying 'ab' by 'ab':

ab * ab = abab [associatively]

= aaba [since bab=aba]

= ba [since aa=1]

Doing this for all the other entries in the Cayley table shows that this completely defines the group. How can we work out the minimum set of relations that will define the group?

Cayley Graph

This shows all the group elements as points with arrows (colour coded) for each

Free Group

This is a group whose presentation has no relations. This means that it is an infinite group (although finitely generated) because each time the generators are applied they will generate new elements of the group. So the relations have the effect of reducing the number of possible elements

Group Examples

So a free group (which nas no relations) is infinite because, each time we apply a generator, a new element is produced. So, in the table below, I have started adding relations to see how many we need to add to make the group finite.

I first added one relation (a²=1), this prevents a sequence of 'a's but we can still have a potentially infinite sequence of 'b's or an alternating 'ababa…' sequence.

I then added (b²=1), but this still allows a potentially infinite alternating sequence 'ababa…'.

It was only when I added a third relation that the group became finite.

  Presentation Cayley graph
Free Group Γ = < a,b… | Ø> free
One relation Γ = < a,b | a² > one
Two relations Γ = < a,b | a²,b² > two
  Γ = < a,b | a²,b²,ab=ba > simple

Dihedral Group

(rotation and reflection of a triangle) see this page for full description.

D3= < a,b | aa,bb,bab=aba>

or

D3= < a,b | aa,bbb,a-1ba=b-1>

cayley graph
Cube (see this page)   cayley graph
complex number C = < a,b | a²=b,b²,ab=ba> I'm not sure if this is minimum representation (this represents a=i, b=-1)
quaternion H = <a,b | a² = b² = (ab)²>
or
H = < a,b | a 4,a²=b²,b-1ab=a-1>
 
Two dimensional Clifford algebra (see this page) Γ= < a,b,-e,-a,-b | a²,b²,(-a)²,(-b)²,(-e)²,a(-a)=-e,b(-b)=-e,ab=-ba > ga2

 

 

 


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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.

flag flag flag flag flag flag Fearless Symmetry - This book approaches symmetry from the point of view of number theory. It may not be for you if you are only interested in the geometrical aspects of symmetry such as rotation groups but if you are interested in subjects like modulo n numbers, Galois theory, Fermats last theorem, to name a few topics the chances are you will find this book interesting. It is written in a friendly style for a general audience but I did not find it dumbed down. I found a lot of new concepts to learn. It certainly gives a flavor of the complexity of the subject and some areas where maths is still being discovered.

 

Terminology and Notation

Specific to this page here:

 

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