Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Cayley Graphs
Free Groups
Examples
Relators
Graphs
Ryan Jensen
Cayley Graphs
F1
University of Tennessee
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
March 26, 2014
Group
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Definition
A group is a nonempty set G with a binary operation ∗ which
satisfies 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 that
a ∗ e = e ∗ a = a for all a ∈ G .
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
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References
(iv) inverse: for each a ∈ G , there is an inverse element
a−1 ∈ G so that a−1 ∗ a = a ∗ a−1 = e.
A group is abelian (or commutative) if a ∗ b = b ∗ a for all
a, 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.
Examples of Groups
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
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Free Groups
Examples
Relators
Graphs
Cayley Graphs
Example: Z
The integers Z = {. . . , −2, −1, 0, 1, 2, . . .} form an abelian
group under the addition operation.
Example: Z2
Define Z/2Z = Z2 = {0̄, 1̄}, where 0̄ = {z ∈ Z | z is even}, and
1̄ = {z ∈ Z | z is odd}. Then Z/2Z is an abelian group.
F1
F2
Presentations
Cayley Color
Graphs
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Example: Zn
Let n ∈ Z, and define Z/nZ = Zn = {0̄, 1̄, . . . n − 1}, where
i¯ = {z ∈ Z | remainder of z|n = i} are known as the integers
modulo n. Then Z/nZ is an abelian group.
A closer look at Z5
Cayley Graphs
Ryan Jensen
Groups
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Cayley Graphs
F1
F2
A multiplication (addition) table is called a Cayley Table. Let’s
look at the Cayley table for the group Z5 = {0, 1, 2, 3, 4}.
∗
0
1
2
3
4
0
0
1
2
3
4
1
1
2
3
4
0
2
2
3
4
0
1
3
3
4
0
1
2
4
4
0
1
2
3
Presentations
Cayley Color
Graphs
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Notice the table is symmetric about the diagonal, meaning the
group is abelian.
Also 1 generates the group, meaning that if we add 1 to itself
enough times, we get the whole group.
Other Examples of Groups
Cayley Graphs
Ryan Jensen
Groups
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Examples
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Cayley Graphs
F1
F2
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Graphs
Examples
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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, is
the group of all n × n invertible matrices with entries in R.
SL(n, R), the special linear group over the real numbers, is
the group of all n × n invertible matrices with entries in R
whose determinant is 1.
GL(2, Z13 ) is the group of 2 × 2 invertible matrices with
entries from Z13 (as before Z13 is a group; it is actually a
field since 13 is prime, but this won’t actually be needed in
this presentation).
Other Examples of Groups
Cayley Graphs
Ryan Jensen
Groups
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Examples
Isomorphisms
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Graphs
Cayley Graphs
F1
F2
Presentations
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Graphs
Examples
Applications
References
Examples of Groups
Sn , the symmetric group on n elements, is the group of
bijections between an n element set and itself.
Dn , the dihedral group of order 2n, is the group of
symmetries of a regular n-gon.
Many others.
Group Isomorphisms
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Definition
Let H and G be groups. A function f : G → H so that
f (ab) = f (a)f (b) for all a, b ∈ G is a homomorphism.
If f is bijective, then f is an isomorphism.
Graphs
If G = H, then f is an automorphism.
Cayley Graphs
If there is an isomorphism between G and G , then G and
H are isomorphic, written G ∼
= H.
F1
F2
Presentations
Cayley Color
Graphs
Examples
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Group isomorphisms are nice since they mean two groups are
the same except for the labeling of their elements.
Subgroups
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Definition
A subset H of a group G is a subgroup if is itself a group under
the operation of G ; that H is a subgroup of G is denoted
H ≤ G.
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
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Graphs
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Definition
If Y is a subset of a group G , then the subset generated by Y
is the collection of all (finite) products of elements of Y . This
subgroup is denoted by hY i. If Y is a finite set with elements
y1 , y2 , . . . yn , then the notation hy1 , y2 , . . . yn i is used. A group
which is generated by a single element is called cyclic.
Examples of Subgroups
Cayley Graphs
Ryan Jensen
Example: Trivial Subgroups
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
For any group G , the group consisting of only the identity is a
subgroup of G , and G is a subgroup of itself.
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
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Graphs
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Example: Even Odd Integers
A somewhat less trivial example is that the even integers are a
subgroup of Z; however, the odd integers are not as there is no
identity element.
Example: nZ
For any integer n ∈ Z, nZ = {nz|z ∈ Z} is a subgroup of Z.
Cartesian Product
Cayley Graphs
Ryan Jensen
Groups
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Forming Groups
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Definition
Let A and B be sets. The Cartesian product of A and B is the
set
A × B = {(a, b) | a ∈ A, b ∈ B}
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
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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)}
Direct Product
Cayley Graphs
Ryan Jensen
Groups
Group Basics
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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 as
the direct product (direct sum if G and H are abelian) of G
and H. The group operation on G × H is done coordinate-wise.
Cayley Graphs
F1
F2
Example: Z2 ⊕ Z3
Presentations
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There is a group of order 6 found by taking the direct sum of
Z2 and Z3 , G = Z2 ⊕ Z3 .
Quotient Groups
Cayley Graphs
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Groups
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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 the
quotient G /H is called the quotient group of G by H, or simply
G mod H.
Cayley Graphs
F1
F2
Presentations
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Example: Z/nZ
Z is a group, and nZ is a normal subgroup of Z. So the
quotient Z/nZ is a group. (Remember Z/nZ = Zn .)
Free Groups
Cayley Graphs
Ryan Jensen
Groups
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Cayley Graphs
F1
F2
Definition
Let A be a set.
The set A = {a1 , a2 , . . .} together with its formal inverses
A−1 = {a1−1 , a2−1 , . . .} 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 its
inverse.
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The collection of all finite reduce words on the alphabet A
is a free group on A, denote by F (A).
The group operation is concatenation of words, followed
by reduction if necessary.
More Notation
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
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Theorem
Let A and B be finite sets, then F (A) is isomorphic to F (B) if
and only if |A| = |B|.
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
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The above Theorem says that only the size of the alphabet is
important when constructing a free group. As a result, when
the alphabet is finite, i.e. |A| = n, the free group on A is
denoted Fn and is called the free group of rank n, or the free
group on n generators.
Examples of Free Groups
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Example: Trivial Free Group
The free group on an empty generating set (or the free group
on 0 generators) is the trivial group consisting of only the
empty word (the identity element).
Free Groups
Examples
Relators
F (∅) = F0 = {e}.
Graphs
Cayley Graphs
F1
F2
Example: F1
Presentations
Cayley Color
Graphs
Examples
Applications
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The free group on one generator is isomorphic to the integers.
F1 = {. . . , a−2 , a−1 , a0 = e, a = a1 , a2 , . . .}
∼ Z by the map ai 7→ i.
F1 =
Examples of Free Groups
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Example: F2
F2 on generators a, b is the collection of all finite words
from the letters a, b, a−1 , b −1 .
Example elements are e = aa−1 , a3 , b −2 , bab −1 .
An example of group operation:
Graphs
Cayley Graphs
(a3 ) ∗ (b −2 ) ∗ (bab −1 ) = a3 b −2 bab −1 = a3 b −1 ab −1
F1
F2
Presentations
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Example: F3
F3 on generators a, b, c is done in a similar manner.
Relators
Cayley Graphs
Ryan Jensen
Definition
Let F be a free group.
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
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Graphs
Examples
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A relator on F is a word defined to be equal to the identity.
For example aba−1 b −1 = e is a relator in F2 .
The least normal subgroup N is the normal subgroup
generated 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 hS | Ri where S is the set of
generators and R the set of relators. hS | Ri is called a
group presentation.
For example h{a, b} | {aba−1 b −1 = ei}, usually
abbreviated ha, b | aba−1 b −1 i.
Examples of Presentations
Cayley Graphs
Example: Trivial Presentations
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
h | i is the trivial group consisting of only the empty word.
ha | i is F1 .
ha, b | i is F2 .
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
Example: Z2
The group given by the presentation ha | a2 i is isomorphic to
Z2 .
The letter a generates F1 = {. . . , a−2 , a−1 , e, a, a2 . . .}.
The relator a2 = e means replace a2 with e for all words in
F1 .
The only elements left are e and a. Hence this group is
isomorphic to Z2 .
A Little Graph Theory
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
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Relators
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 0 , v 0 ) if and only if either
1
u = u 0 and vv 0 ∈ E (H), or
2
v = v 0 and uu 0 ∈ E (G ).
Graphs
Cayley Graphs
F1
F2
Definition (From West)
Presentations
Cayley Color
Graphs
Examples
Applications
References
A directed graph or digraph G is a triple consisting of a vertex
set V (G ), and edge set E (G ), and a function assigning each
edge an ordered pair of vertices. The first vertex of the ordered
pair is the tail of the edge, and the second is the head; together
they are the endpoints.
Cartesian Product of Graphs
Cayley Graphs
a
1
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
2
c
b
Free Groups
(2, a)
Examples
Relators
Graphs
Cayley Graphs
F1
(1, a)
F2
Presentations
Cayley Color
Graphs
Examples
(1, b)
Applications
(1, c)
References
(2, b)
(2, c)
Directed Graphs
Cayley Graphs
Directed graphs just have directed edges.
Ryan Jensen
a
1
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
2
c
b
Free Groups
Examples
Relators
(2, a)
Graphs
Cayley Graphs
F1
F2
(1, a)
Presentations
Cayley Color
Graphs
Examples
Applications
(1, b)
(1, c)
References
(2, b)
(2, c)
Cayley Graphs
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Definition
Let Γ be a group with generating set S. The Cayley Graph of Γ
with respect to S, denoted ∆ = ∆(Γ; S), is the graph with
V (∆) = Γ, and an edge between vertices g , h ∈ Γ if
g −1 h ∈ S ∪ S −1 .
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
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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 Graph of F1
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
We will construct a Cayley Graph for F1 ∼
= Z.
−2
−1
Recall that F1 = ha | i = {. . . a , a , a0 , a1 , a2 , . . .}.
We want to draw ∆ = ∆(F1 , a) = ∆(ha | i).
The vertices of ∆ are the elements of F1 .
Take any element ai ∈ F1 , since a is a generator, there is
an edge between ai and ai a = ai+1 .
The result is an infinite graph, the real line R.
a−2
a−1
a0
a1
a2
Another Cayley Graph of F1
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
The Cayley Graph depends both on the group and on the
generating set chosen. Lets look at ∆(ha, a2 | i).
Free Groups
a−1
Examples
Relators
a1
a3
a5
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
a−2
a0
a2
a4
a6
Yet Another Cayley Graph of F1
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Lets draw ∆(ha2 , a3 | i).
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
a−4
a−3
a−2
a−1
a0
a1
a2
a3
a4
Canonical Cayley Graph of F2
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
Lets look at ∆(ha, b | i).
Canonical Cayley Graph of F2
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
Examples of how to draw Cayley Graphs
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Lets draw ha | a2 i.
Free Groups
Examples
Relators
a−4
a−3
a−2
a−1
a0
a1
a0
a1
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
a2
a3
a4
Examples of how to draw Cayley Graphs
Cayley Graphs
Ryan Jensen
Now lets do ha | a3 i.
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
a−4
a−3
a−2
a−1
a0
a1
Examples
Relators
Graphs
Cayley Graphs
F1
F2
a2
Presentations
Cayley Color
Graphs
Examples
Applications
References
a0
a1
a2
a3
a4
Examples of how to draw Cayley Graphs
Cayley Graphs
ha | a5 i, in a different approach.
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
a0
a1
a2
a3
a4
Free Groups
Examples
Relators
a1
Graphs
Cayley Graphs
a2
F1
F2
Presentations
a0
Cayley Color
Graphs
Examples
Applications
a3
References
a4
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
From these examples, we can see that:
ha | an i = {a0 , a1 , . . . an−1 }, and generated by a.
Zn = {0, 1, . . . n − 1}, and generated by 1.
So Zn ∼
= ha | an i by the map i 7→ ai .
Zn is known as the cyclic group of order n, and the Cayley
graph is the cyclic graph of length n.
Examples of how to draw Cayley Graphs
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
∆(ha, b | aba−1 b −1 i)
aba−1 b −1 = e if and only if ab = ba.
Facts about Cayley Graphs
Cayley Graphs
Facts
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
The degree of each vertex is equal to the total number of
generators, i.e. |S ∪ S −1 |.
Relators in a group presentation correspond to cycles in
the Cayley Graph.
A group is abelian if and only if for each pair of generators
a, b, the path aba−1 b −1 is closed.
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
The Cayley Graph of a group depends on the group, and
the group presentation.
A Cayley Graph exists for each finite group (each finite
group has a finite presentation).
Subgroups of a group can be found by looking at
sub-graphs generated by elements of the group.
Cayley Color Graphs
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
A Cayley Color Graph is the same as a Cayley Graph, except we
no longer include the inverses of generating elements by default.
Definition
Let Γ be a group with generating set S. The Cayley Color
Graph of Γ with respect to S, denoted ∆C = ∆C (Γ; S), is the
graph with V (∆) = Γ, and an edge between vertices g , h ∈ Γ if
g −1 h ∈ S.
F1
F2
Presentations
So ∆C (ha | i) is
Cayley Color
Graphs
Examples
Applications
References
a−2
a−1
a0
a1
a2
Examples of Cayley Color Graphs
Cayley Graphs
Ryan Jensen
∆C (ha | a5 i).
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
a1
a2
Examples
Relators
Graphs
a0
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
a3
a4
Examples of Cayley Color Graphs
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Lets look at the Cayley Color Graph of ha, b | a2 , abab, b 3 i,
which is a presentation for the group D3 . First notice that
a2 = e means a = a−1 , b 3 = e means b −1 = b 2 .
abab = e iff aba = b −1 iff aba = b 2 .
From above, b = (aa)b(aa) = a(aba)a = ab 2 a.
Free Groups
Examples
Relators
b2
ab 2
b
ab
e
a
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
Examples of Cayley Color Graphs
Cayley Graphs
Ryan Jensen
We can redraw the graph.
Groups
a
(2, a)
e
(1, a)
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
b
F1
b2
(1, b)
(1, c)
F2
Presentations
Cayley Color
Graphs
ab 2
ab
(2, b)
(2, c)
Examples
Applications
References
Now we can compare it to a graph we have already seen, the
Cartesian product of a directed path on two vertices with a
directed 3 cycle.
Theorems
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Theorem
Let ∆C (hS | Ri) be a Cayley Color graph for a finite group.
Then Aut (hS | Ri) ∼
= hS | Ri, this is not dependent on the
presentation of the group.
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
Corollary
If ∆C (hS1 | R1 i) ∼
= ∆C (hS2 | R2 i), then hS1 | R1 i ∼
= hS2 | R2 i.
Theorems
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
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 ).
Free Groups
Examples
Relators
Specifically,
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
∆C (hs1 , . . . , sm | r1 , . . . rt i) ∆C (hsm+1 , . . . , sn | rt+1 , . . . rq i)
= ∆C (hs1 , . . . sn | r1 , . . . , rq , si sj si−1 sj−1 i)
for all 1 ≤ i ≤ m ≤ j ≤ n.
All this Theorem is saying is that the product of groups and the
product of their respective Cayley color graphs behave in a nice
way. We won’t worry too much about the presentations.
Examples of Cayley Color Graphs
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Using the previous Theorem, we can find the standard Cayley
color graph of Z2 ⊕ Z3 (Z2 = ha | a2 i, and Z3 = ha | a3 i).
∆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 ).
Free Groups
Examples
Relators
(1, 0)
Graphs
Cayley Graphs
F1
F2
(0, 0)
Presentations
Cayley Color
Graphs
Examples
(0, 1)
Applications
(0, 2)
References
(1, 1)
(1, 2)
Examples of Cayley Color Graphs
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
The Cayley of S3 = ha, b | a2 , b 2 , (ab)3 i is shown below. Note
that S3 is usually written
S3 = {(), (12), (13), (23), (123), (132)}, the vertices are labeled
this way.
Free Groups
(12)
Examples
Relators
Graphs
Cayley Graphs
()
F1
F2
Presentations
Cayley Color
Graphs
(123)
Examples
(132)
Applications
References
(23)
(13)
Examples of Cayley Color Graphs
Cayley Graphs
Ryan Jensen
We can now analyze the groups D3 and S3 .
Groups
a
(12)
e
()
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
b
F1
b2
(123)
(132)
F2
Presentations
Cayley Color
Graphs
ab 2
ab
(23)
(13)
Examples
Applications
References
Since they Cayley color graphs are isomorphic, the groups D3
and S3 are isomorphic, even though they may not have the
same presentation.
Examples of Cayley Color Graphs
Cayley Graphs
Ryan Jensen
Now lets look at the groups D3 and Z2 ⊕ Z3 .
Groups
a
(1, 0)
e
(0, 0)
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
b
F2
b2
(0, 1)
(0, 2)
Presentations
Cayley Color
Graphs
ab 2
ab
(1, 1)
(1, 2)
Examples
Applications
References
These groups are not isomorphic, as D3 is not abelian, and
Z2 ⊕ Z3 is.
Applications in Math
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Theorem
A subgroup of a free group is a free group.
Proof (Basic Idea)
Free Groups
Examples
Relators
Let F be a free group, and G a subgroup of F .
Graphs
1
F is free of relators.
Cayley Graphs
2
The Cayley graph ∆(F ) is a tree (no cycles).
Presentations
3
The Cayley graph ∆(G ) is a connected sub-graph of ∆(F ).
Cayley Color
Graphs
4
So ∆(G ) is a tree.
5
So the presentation of G is free of relators.
6
Hence G is a free group.
F1
F2
Examples
Applications
References
Applications in Math
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
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 sense
the 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 Z
eventually looks like the real number line.
We look at the ends of spaces, i.e. ends of a space
quasi-isometric to F2 .
Applications in Computer Science
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
Presentations
Cayley Color
Graphs
Examples
Applications
References
Langston et al.
Application was in parallel processing.
Problem was to create large graphs of given degree and
diameter.
Approach was to use Cayley graphs as the underlying
group controls the degree, and the diameter is easy (since
Cayley graphs are vertex transitive).
Several records where broken for the largest graph of given
degree and diameter.
Applications in Computer Science
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
Free Groups
Examples
Relators
Graphs
Cayley Graphs
F1
F2
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 all
elements with determinant of ±1.
The generators where
0 1
11 2
11 4
order 2,
order 52,
order 14.
1 0
8 12
7 5
Presentations
Cayley Color
Graphs
Examples
The Cayley graph of this group has degree 5, diameter 7,
and has 4368 vertices.
Applications
References
A new record.
References
Cayley Graphs
Ryan Jensen
Groups
Group Basics
Examples
Isomorphisms
Forming Groups
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 diameter
symmetric 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.
Free Groups
Examples
Relators
Graphs
Thomas W Hungerford, Algebra, volume 73 of graduate texts in mathematics, Springer-Verlag, New
York, 1980.
Bernard Knueven, Graph automorphisms, 2014.
Cayley Graphs
F1
F2
Serge Lang, Algebra revised third edition, Springer-Verlag, 2002.
Presentations
James Munkres, Topology (2nd edition), 2 ed., Pearson, 2000.
Cayley Color
Graphs
Piotr W Nowak and Guoliang Yu, Large scale geometry, 2012.
Examples
Douglas B. West, Introduction to graph theory (2nd edition), 2 ed., Pearson, 2000.
Applications
References
A.T. White, Graphs of groups on surfaces, volume 188: Interactions and models (north-holland
mathematics studies), 1 ed., North Holland, 5 2001.