Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Cayley Graphs

Ryan Jensen

University of Tennessee

March 26, 2014

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Group

Definition

A group is a nonempty set G with a binary operation ∗ whichsatisfies the following:

(i) closure: if a, b ∈ G , then a ∗ b ∈ G .

(ii) associative: a ∗ (b ∗ c) = (a ∗ b) ∗ c for all a, b, c ∈ G .

(iii) identity: there is an identity element e ∈ G so thata ∗ e = e ∗ a = a for all a ∈ G .

(iv) inverse: for each a ∈ G , there is an inverse elementa−1 ∈ G so that a−1 ∗ a = a ∗ a−1 = e.

A group is abelian (or commutative) if a ∗ b = b ∗ a for alla, b ∈ G .

We usually write ab in place of a ∗ b if the operation is known.When the group is abelian, we write a + b.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Groups

Example: Z

The integers Z = {. . . ,−2,−1, 0, 1, 2, . . .} form an abeliangroup under the addition operation.

Example: Z2

Define Z/2Z = Z2 = {0, 1}, where 0 = {z ∈ Z | z is even}, and1 = {z ∈ Z | z is odd}. Then Z/2Z is an abelian group.

Example: Zn

Let n ∈ Z, and define Z/nZ = Zn = {0, 1, . . . n − 1}, wherei = {z ∈ Z | remainder of z |n = i} are known as the integersmodulo n. Then Z/nZ is an abelian group.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

A closer look at Z5

A multiplication (addition) table is called a Cayley Table. Let’slook at the Cayley table for the group Z5 = {0, 1, 2, 3, 4}.

∗ 0 1 2 3 4

0 0 1 2 3 41 1 2 3 4 02 2 3 4 0 13 3 4 0 1 24 4 0 1 2 3

Notice the table is symmetric about the diagonal, meaning thegroup is abelian.Also 1 generates the group, meaning that if we add 1 to itselfenough times, we get the whole group.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Other Examples of Groups

There are many examples of groups, here are a few more:

Examples of Groups

GL(n,R), the general linear group over the real numbers, isthe group of all n× n invertible matrices with entries in R.

SL(n,R), the special linear group over the real numbers, isthe group of all n × n invertible matrices with entries in Rwhose determinant is 1.

GL(2,Z13) is the group of 2× 2 invertible matrices withentries from Z13 (as before Z13 is a group; it is actually afield since 13 is prime, but this won’t actually be needed inthis presentation).

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Other Examples of Groups

Examples of Groups

Sn, the symmetric group on n elements, is the group ofbijections between an n element set and itself.

Dn, the dihedral group of order 2n, is the group ofsymmetries of a regular n-gon.

Many others.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Group Isomorphisms

Definition

Let H and G be groups. A function f : G → H so thatf (ab) = f (a)f (b) for all a, b ∈ G is a homomorphism.

If f is bijective, then f is an isomorphism.

If G = H, then f is an automorphism.

If there is an isomorphism between G and G , then G andH are isomorphic, written G ∼= H.

Group isomorphisms are nice since they mean two groups arethe same except for the labeling of their elements.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Subgroups

Definition

A subset H of a group G is a subgroup if is itself a group underthe operation of G ; that H is a subgroup of G is denotedH ≤ G .

Definition

If Y is a subset of a group G , then the subset generated by Yis the collection of all (finite) products of elements of Y . Thissubgroup is denoted by 〈Y 〉. If Y is a finite set with elementsy1, y2, . . . yn, then the notation 〈y1, y2, . . . yn〉 is used. A groupwhich is generated by a single element is called cyclic.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Subgroups

Example: Trivial Subgroups

For any group G , the group consisting of only the identity is asubgroup of G , and G is a subgroup of itself.

Example: Even Odd Integers

A somewhat less trivial example is that the even integers are asubgroup of Z; however, the odd integers are not as there is noidentity element.

Example: nZ

For any integer n ∈ Z, nZ = {nz |z ∈ Z} is a subgroup of Z.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Cartesian Product

Definition

Let A and B be sets. The Cartesian product of A and B is theset

A× B = {(a, b) | a ∈ A, b ∈ B}

Example

Let A = {1, 2} and B = {a, b, c} then

A× B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Direct Product

Definition

Given two groups G and H, their Cartesian product G × H,(denoted G ⊕ H if G and H are abelian) is a group known asthe direct product (direct sum if G and H are abelian) of Gand H. The group operation on G ×H is done coordinate-wise.

Example: Z2 ⊕ Z3

There is a group of order 6 found by taking the direct sum ofZ2 and Z3, G = Z2 ⊕ Z3.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Quotient Groups

Without going into too many technicalities about cosets,normal subgroups etc., quotient groups can be defined.

Definition

Let G be a group and H a normal subgroup of G . Then thequotient G/H is called the quotient group of G by H, or simplyG mod H.

Example: Z/nZ

Z is a group, and nZ is a normal subgroup of Z. So thequotient Z/nZ is a group. (Remember Z/nZ = Zn.)

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Free Groups

Definition

Let A be a set.

The set A = {a1, a2, . . .} together with its formal inversesA−1 = {a−11 , a−12 , . . .} from an alphabet.

The elements of A ∪ A−1 are called letters.

A word is a concatenation of letters.

A reduced word is a word where no letter is adjacent to itsinverse.

The collection of all finite reduce words on the alphabet Ais a free group on A, denote by F (A).

The group operation is concatenation of words, followedby reduction if necessary.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

More Notation

Theorem

Let A and B be finite sets, then F (A) is isomorphic to F (B) ifand only if |A| = |B|.

The above Theorem says that only the size of the alphabet isimportant when constructing a free group. As a result, whenthe alphabet is finite, i.e. |A| = n, the free group on A isdenoted Fn and is called the free group of rank n, or the freegroup on n generators.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Free Groups

Example: Trivial Free Group

The free group on an empty generating set (or the free groupon 0 generators) is the trivial group consisting of only theempty word (the identity element).

F (∅) = F0 = {e}.

Example: F1

The free group on one generator is isomorphic to the integers.

F1 = {. . . , a−2, a−1, a0 = e, a = a1, a2, . . .}F1 ∼= Z by the map ai 7→ i .

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Free Groups

Example: F2

F2 on generators a, b is the collection of all finite wordsfrom the letters a, b, a−1, b−1.

Example elements are e = aa−1, a3, b−2, bab−1.

An example of group operation:

(a3) ∗ (b−2) ∗ (bab−1) = a3b−2bab−1 = a3b−1ab−1

Example: F3

F3 on generators a, b, c is done in a similar manner.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Relators

Definition

Let F be a free group.

A relator on F is a word defined to be equal to the identity.

For example aba−1b−1 = e is a relator in F2.

The least normal subgroup N is the normal subgroupgenerated by a set of relators.

A new group is formed by taking the quotient of F by N,F/N.

Compact notation for this is 〈S |R〉 where S is the set ofgenerators and R the set of relators. 〈S |R〉 is called agroup presentation.

For example 〈{a, b} | {aba−1b−1 = e〉}, usuallyabbreviated 〈a, b | aba−1b−1〉.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Presentations

Example: Trivial Presentations

〈 | 〉 is the trivial group consisting of only the empty word.

〈a | 〉 is F1.

〈a, b | 〉 is F2.

Example: Z2

The group given by the presentation 〈a | a2〉 is isomorphic toZ2.

The letter a generates F1 = {. . . , a−2, a−1, e, a, a2 . . .}.The relator a2 = e means replace a2 with e for all words inF1.

The only elements left are e and a. Hence this group isisomorphic to Z2.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

A Little Graph Theory

Definition (From West)

The Cartesian product of two graphs G and H, written G � H,is the graph with vertex set V (G )× V (H) specified by putting(u, v) adjacent to (u′, v ′) if and only if either

1 u = u′ and vv ′ ∈ E (H), or

2 v = v ′ and uu′ ∈ E (G ).

Definition (From West)

A directed graph or digraph G is a triple consisting of a vertexset V (G ), and edge set E (G ), and a function assigning eachedge an ordered pair of vertices. The first vertex of the orderedpair is the tail of the edge, and the second is the head; togetherthey are the endpoints.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Cartesian Product of Graphs

1

2

a

b c

(1, a)

(1, b) (1, c)

(2, a)

(2, b) (2, c)

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Directed Graphs

Directed graphs just have directed edges.

1

2

a

b c

(1, a)

(1, b) (1, c)

(2, a)

(2, b) (2, c)

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Cayley Graphs

Definition

Let Γ be a group with generating set S . The Cayley Graph of Γwith respect to S , denoted ∆ = ∆(Γ; S), is the graph withV (∆) = Γ, and an edge between vertices g , h ∈ Γ ifg−1h ∈ S ∪ S−1.

Another way to think of the edges is if g ∈ Γ and s ∈ S ∪ S−1,then there is an edge connecting g and gs.This becomes easier to see with some examples.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Cayley Graph of F1

We will construct a Cayley Graph for F1 ∼= Z.

Recall that F1 = 〈a | 〉 = {. . . a−2, a−1, a0, a1, a2, . . .}.We want to draw ∆ = ∆(F1, a) = ∆(〈a | 〉).

The vertices of ∆ are the elements of F1.

Take any element ai ∈ F1, since a is a generator, there isan edge between ai and aia = ai+1.

The result is an infinite graph, the real line R.

a−2 a−1 a0 a1 a2

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Another Cayley Graph of F1

The Cayley Graph depends both on the group and on thegenerating set chosen. Lets look at ∆(〈a, a2 | 〉).

a−2

a−1

a0

a1

a2

a3

a4

a5

a6

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Yet Another Cayley Graph of F1

Lets draw ∆(〈a2, a3 | 〉).

a0 a2 a3 a4a1a−4 a−3 a−2 a−1 a0 a1 a2 a3 a4

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Canonical Cayley Graph of F2

Lets look at ∆(〈a, b | 〉).

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Canonical Cayley Graph of F2

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of how to draw Cayley Graphs

Lets draw 〈a | a2〉.

a−4 a−3 a−2 a−1 a0 a1 a2 a3 a4

a0 a1

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of how to draw Cayley Graphs

Now lets do 〈a | a3〉.

a−4 a−3 a−2 a−1 a0 a1 a2 a3 a4

a0 a1

a2

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of how to draw Cayley Graphs

〈a | a5〉, in a different approach.

a0 a1 a2 a3 a4

a0

a1

a2

a3

a4

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

From these examples, we can see that:

〈a | an〉 = {a0, a1, . . . an−1}, and generated by a.

Zn = {0, 1, . . . n − 1}, and generated by 1.

So Zn∼= 〈a | an〉 by the map i 7→ ai .

Zn is known as the cyclic group of order n, and the Cayleygraph is the cyclic graph of length n.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of how to draw Cayley Graphs

∆(〈a, b | aba−1b−1〉)aba−1b−1 = e if and only if ab = ba.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Facts about Cayley Graphs

Facts

The degree of each vertex is equal to the total number ofgenerators, i.e. |S ∪ S−1|.Relators in a group presentation correspond to cycles inthe Cayley Graph.

A group is abelian if and only if for each pair of generatorsa, b, the path aba−1b−1 is closed.

The Cayley Graph of a group depends on the group, andthe group presentation.

A Cayley Graph exists for each finite group (each finitegroup has a finite presentation).

Subgroups of a group can be found by looking atsub-graphs generated by elements of the group.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Cayley Color Graphs

A Cayley Color Graph is the same as a Cayley Graph, except weno longer include the inverses of generating elements by default.

Definition

Let Γ be a group with generating set S . The Cayley ColorGraph of Γ with respect to S , denoted ∆C = ∆C (Γ;S), is thegraph with V (∆) = Γ, and an edge between vertices g , h ∈ Γ ifg−1h ∈ S .

So ∆C (〈a | 〉) is

a−2 a−1 a0 a1 a2

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Cayley Color Graphs

∆C (〈a | a5〉).

a0

a1

a2

a3

a4

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Cayley Color Graphs

Lets look at the Cayley Color Graph of 〈a, b | a2, abab, b3〉,which is a presentation for the group D3. First notice that

a2 = e means a = a−1, b3 = e means b−1 = b2.

abab = e iff aba = b−1 iff aba = b2.

From above, b = (aa)b(aa) = a(aba)a = ab2a.

e a

b

b2

ab

ab2

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Cayley Color Graphs

We can redraw the graph.

e

b b2

a

ab2 ab

(1, a)

(1, b) (1, c)

(2, a)

(2, b) (2, c)

Now we can compare it to a graph we have already seen, theCartesian product of a directed path on two vertices with adirected 3 cycle.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Theorems

Theorem

Let ∆C (〈S |R〉) be a Cayley Color graph for a finite group.Then Aut (〈S |R〉) ∼= 〈S |R〉, this is not dependent on thepresentation of the group.

Corollary

If ∆C (〈S1 |R1〉) ∼= ∆C (〈S2 |R2〉), then 〈S1 |R1〉 ∼= 〈S2 |R2〉.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Theorems

Theorem

Let H and G be finite groups with presentations PH and PG .Then there is a presentation for H × G so that

∆C (PH) � ∆C (PG ) ∼= ∆C (PH × PG ).

Specifically,

∆C (〈s1, . . . , sm | r1, . . . rt〉) � ∆C (〈sm+1, . . . , sn | rt+1, . . . rq〉)= ∆C (〈s1, . . . sn | r1, . . . , rq, si sjs−1i s−1j 〉)

for all 1 ≤ i ≤ m ≤ j ≤ n.

All this Theorem is saying is that the product of groups and theproduct of their respective Cayley color graphs behave in a niceway. We won’t worry too much about the presentations.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Cayley Color Graphs

Using the previous Theorem, we can find the standard Cayleycolor graph of Z2 ⊕ Z3 (Z2 = 〈a | a2〉, and Z3 = 〈a | a3〉).

∆C (Z2) is a directed cycle of size 2

∆C (Z3) is a directed cycle of size 3.

Hence (standard) ∆C (Z2 ⊕ Z3) = ∆C (Z2) � ∆C (Z3).

(0, 0)

(0, 1) (0, 2)

(1, 0)

(1, 1) (1, 2)

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Cayley Color Graphs

The Cayley of S3 = 〈a, b | a2, b2, (ab)3〉 is shown below. Notethat S3 is usually writtenS3 = {(), (12), (13), (23), (123), (132)}, the vertices are labeledthis way.

()

(123) (132)

(12)

(23) (13)

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Cayley Color Graphs

We can now analyze the groups D3 and S3.

e

b b2

a

ab2 ab

()

(123) (132)

(12)

(23) (13)

Since they Cayley color graphs are isomorphic, the groups D3

and S3 are isomorphic, even though they may not have thesame presentation.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Examples of Cayley Color Graphs

Now lets look at the groups D3 and Z2 ⊕ Z3.

e

b b2

a

ab2 ab

(0, 0)

(0, 1) (0, 2)

(1, 0)

(1, 1) (1, 2)

These groups are not isomorphic, as D3 is not abelian, andZ2 ⊕ Z3 is.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Applications in Math

Theorem

A subgroup of a free group is a free group.

Proof (Basic Idea)

Let F be a free group, and G a subgroup of F .

1 F is free of relators.

2 The Cayley graph ∆(F ) is a tree (no cycles).

3 The Cayley graph ∆(G ) is a connected sub-graph of ∆(F ).

4 So ∆(G ) is a tree.

5 So the presentation of G is free of relators.

6 Hence G is a free group.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Applications in Math

What I use Cayley graphs for: Large Scale Geometry.

Take an arbitrary space (topological, geometrical etc.).

Take the Cayley graph of a group.

Look at both from far away.

If they look the same (quasi-isometric), then in some sensethe space has the group inside it.

Some interesting things about Large Scale Geometry.

We are not concerned about small things.

So any finite graph is trivial, as it becomes a point.

So we only work with infinite graphs.

Example: Any Cayley graph of a presentation of Zeventually looks like the real number line.

We look at the ends of spaces, i.e. ends of a spacequasi-isometric to F2.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Applications in Computer Science

Langston et al.

Application was in parallel processing.

Problem was to create large graphs of given degree anddiameter.

Approach was to use Cayley graphs as the underlyinggroup controls the degree, and the diameter is easy (sinceCayley graphs are vertex transitive).

Several records where broken for the largest graph of givendegree and diameter.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

Applications in Computer Science

Here is an example group/Cayley graph from their paper.

Example from Langston et al.

The group was a subgroup of GL(2,Z13) consisting of allelements with determinant of ±1.

The generators where[0 11 0

]order 2,

[11 28 12

]order 52,

[11 47 5

]order 14.

The Cayley graph of this group has degree 5, diameter 7,and has 4368 vertices.

A new record.

Cayley Graphs

Ryan Jensen

Groups

Group Basics

Examples

Isomorphisms

Forming Groups

Free Groups

Examples

Relators

Graphs

Cayley Graphs

F1F2Presentations

Cayley ColorGraphs

Examples

Applications

References

References

Brian H. Bowditch, A course on geometric group theory, MSJ Memoirs, vol. 16, Mathematical

Society of Japan, Tokyo, 2006. MR 2243589 (2007e:20085)

Lowell Campbell, Gunnar E. Carlsson, Michael J. Dinneen, Vance Faber, Michael R. Fellows,

Michael A. Langston, James W. Moore, Andrew P. Mullhaupt, and Harlan B. Sexton, Small diametersymmetric networks from linear groups, IEEE Transactions on Computers 41 (1992), no. 2, 218–220.

David S Dummit and Richard M Foote, Abstract algebra, (2004), John Wiley and Sons, Inc.

Thomas W Hungerford, Algebra, volume 73 of graduate texts in mathematics, Springer-Verlag, New

York, 1980.

Bernard Knueven, Graph automorphisms, 2014.

Serge Lang, Algebra revised third edition, Springer-Verlag, 2002.

James Munkres, Topology (2nd edition), 2 ed., Pearson, 2000.

Piotr W Nowak and Guoliang Yu, Large scale geometry, 2012.

Douglas B. West, Introduction to graph theory (2nd edition), 2 ed., Pearson, 2000.

A.T. White, Graphs of groups on surfaces, volume 188: Interactions and models (north-holland

mathematics studies), 1 ed., North Holland, 5 2001.