CAYLEY GRAPHS
RYAN JENSEN
Abstract. These notes are to help with the Homework. Not everything is as mathematically precise as
one would find in a textbook.
1. Groups
The information in this section is well know and can be found in most introductory books on group theory.
See for example [4], [6], or [3].
1.1. Basic Examples of Groups.
Definition 1. 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.
(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.
In practice one usually writes ab in place of a ∗ b, unless one wishes to emphasize the operation. Also if
the group is known to be abelian, the notation a + b is used instead.
Example 1. The integers Z = {. . . , −2, −1, 0, 1, 2, . . .} form an abelian group under the addition operation.
Proof. For any two integers m, n ∈ Z, their sum m + n is also in Z.Clearly the integers are associative, the
identity element is 0, the inverse of n is −n, and commutativity should be clear.
Example 2. Define Z/2Z = Z = {0̄, 1̄}, where 0̄ = {z ∈ Z | z is even}, and 1̄ = {z ∈ Z | z is odd}. Then
Z/2Z is an abelian group.
Proof. It is strait forward to check that Z/2Z satisfies the group axioms. The set is closed under addition
since the sum of any two integers is either even or odd, associativity follows from the associativity of the
integers, as does commutativity. The identity element is 0̄, each element is its own inverse.
Notice that 0̄ and 1̄ could also be defined to be those integers whose remainder when divided by 2 is
respectively 0 and 1. The group Z/2Z is known as the integers modulo 2.
Example 3. Let n ∈ Z, and define Z/nZ = Zn = {0̄, 1̄, . . . n − 1}, where ī = {z ∈ Z | i = remainder of z|n}
are known as the integers modulo n. Then Z/nZ is an abelian group.
Proof. Similar arguments to those given in the proof that Z/2/Z is a group.
Definition 2. The integers modulo n, Z/nZ, are called the cyclic group of order n. The integers, Z, are
known as the infinite cyclic group.
Definition 3. The symmetric group, Sn , is the set of all permutations (i.e bijections) from a set of n elements
to itself. That is if [n] = {1, 2, . . . n}, then
(1)
Sn = {τ : [n] → [n] | τ is a bijection}.
There are n! bijections on a set of size n, so |Sn | = n!. It is usually easiest to use cycle notation when
talking about symmetric groups. In his presentation on Graph Automorphisms Bernard used cycle notation,
so it should be familiar [5]. As a refresher, (1, 4, 5)(3, 7) ∈ S7 is the permutation which does the following:
1 7→ 4, 4 7→ 5, 5 7→ 1, 3 7→ 7, 7 7→ 3, 2 7→ 2, and 6 7→ 6.
1
2
RYAN JENSEN
Example 4. Figure 1 is a Cayley Table of the group S3 , which is simply a multiplication table for the group.
From the Cayley table, it is strait forward to verify that the group axioms are satisfied for S3 .
∗
()
(2, 3)
(1, 2)
(1, 2, 3)
(1, 3, 2)
(1, 3)
()
()
(2, 3)
(1, 2)
(1, 2, 3)
(1, 3, 2)
(1, 3)
(2, 3)
(2, 3)
()
(1, 3, 2)
(1, 3)
(1, 2)
(1, 2, 3)
(1, 2)
(1, 2)
(1, 2, 3)
()
(2, 3)
(1, 3)
(1, 3, 2)
(1, 2, 3)
(1, 2, 3)
(1, 2)
(1, 3)
(1, 3, 2)
()
(2, 3)
(1, 3, 2)
(1, 3, 2)
(1, 3)
(2, 3)
()
(1, 2, 3)
(1, 2)
(1, 3)
(1, 3)
(1, 3, 2)
(1, 2, 3)
(1, 2)
(2, 3)
()
Figure 1. Cayley Table for S3
Example 5. Figure 2 is a Cayley table of the group Z5 . From the table it can be seen that 1 generates the
group, and the group is abelian since the table is symmetric about the origin.
∗
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
Figure 2. Cayley Table for Z5
Example 6. There are many examples of groups, here are a few more:
• 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, Z/13Z) is the group of 2×2 invertible matrices with entries from Z/13Z (by Example 3, Z/13Z
is a group; it is actually a field since 13 is prime, but this won’t actually be needed in this paper).
• Dn , the dihedral group of order 2n, is the group of symmetries of a regular n-gon.
• Many others.
1.2. Subgroups, Homomorphisms, Products, and Quotients.
Definition 4. 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.
Example 7. For any group G, the group consisting of only the identity is a subgroup of G, and G is a
subgroup of itself.
Example 8. 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.
For any integer n ∈ Z, nZ = {nz|z ∈ Z} is a subgroup of Z.
An easy way to get a subgroup from a group G is to take a collection of elements of G, and then consider
all possible products of those elements. This will be a subgroup of G.
Definition 5. 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.
Definition 6. 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. If G = H, then f is an automorphism.
CAYLEY GRAPHS
3
Group isomorphisms are nice since they mean two groups are the same except for the labeling of their
elements. For example, any infinite cyclic group is isomorphic to the integers (simply map the identity to 1),
many properties of the integers carry over to any cyclic group. In a similar manner, any finite cyclic group
of order n is isomorphic to Z/nZ.
Considering only sets for a moment, a very common way of obtaining new sets is via the Cartesian product
of two (or more) sets.
Definition 7. Let A and B be sets. The Cartesian product of A and B is the set
(2)
A × B = {(a, b) | a ∈ A, b ∈ B}
Definition 8. 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.
Example 9. Recall that Zn = Z/nZ. There is a group of order 6 found by taking the direct sum of Z2 and
Z3 , G = Z2 ⊕ Z3 . It can be shown that G is isomorphic to S3 . G is also the smallest group which is not
abelian.
As above, new groups can be formed by taking the product of existing groups. In a similar manner, a new
group can be formed by ”dividing” two groups. Without going into too many technicalities about cosets,
normal subgroups etc., quotient groups can be defined.
Definition 9. 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.
Example 10. Z is a group, and nZ is a normal subgroup of Z. So the quotient Z/nZ is a group, as shown
in Example 3.
1.3. Free Groups. There are several equivalent definitions of a free group. Most require more mathematical
machinery than that which is present in this paper. However, an intuitive approach on the construction of
free groups is given, and the meaning of a free group should be clear after some examples. The details of
this construction can be found in [7] or [1], or as a more algebraic approach in [4].
Let A be a set, known as the alphabet. From the letters (elements) of the alphabet, and the inverses of
letters in the alphabet, words may be formed by concatenation. A word is reduced if no letter is adjacent to
its inverse. The collection of all finite reduced words is a free group on the alphabet A, denoted by F (A).
The group operation is concatenation (followed by reduction if possible). Notice that the empty word is the
identity element.
Theorem 1. Let A and B be finite sets, then F (A) is isomorphic to F (B) if and only if |A| = |B|.
Theorem 1 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.
Example 11. 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).
(3)
F (∅) = F0 = {e}.
Example 12. The free group on one generator is isomorphic to the integers.
Proof. Suppose that {a} is our generating set. Then F1 = F ({a}) = the set of all finite reduced words
using the letters a and a−1 . For example the following are some elements of F1 : a, a−1 , e = aa−1 , aaaa =
a4 , a−1 a−1 a−1 = a−3 . In fact
F1 = {. . . , a−2 , a−1 , a0 = e, a = a1 , a2 , . . .},
and the isomorphism between F1 and Z is ai 7→ i for all i.
Example 13. The free group on two generators a and b is the collection of all finite words from the letters
a, b, a−1 , b−1 . Some example elements of F2 are: e = aa−1 , a3 , b−2 , bab−1 . Again the group operation is
concatenation of words followed by reduction, so (a3 ) ∗ (b−2 ) ∗ (bab−1 ) = a3 b−2 bab−1 = a3 b−1 ab−1 .
4
RYAN JENSEN
a
1
2
c
b
(2, a)
(1, a)
(1, b)
(1, c)
(2, b)
(2, c)
Figure 3. The graph Cartesian product of P2 and C3 .
A relator on a free group is a word in the free group which is set equal to the identity. A new group
is formed by taking the quotient of a free group by the normal group generated by a set of relators. The
notaion for this is G = hA | Ri, where A is the alphabet generating the free group, and R is the set of relators;
hA | Ri is called the presentation of the group G.
Example 14. Let G = ha | i, then G is isomorphic to F1 (and hence isomorphic to Z). Likewise ha, b | i is
isomorphic to F2 .
Proof. These can both be seeing by noting that there are no relators, hence the result is a free group. The
name free group comes from the fact that the group is free of relators.
Example 15. Let G = ha | a2 = ei (usually written simply as ha | a2 i), then G is isomorphic to Z/2Z.
Proof. The letter a generates F1 = {. . . , a−2 , a−1 , e, a, a2 . . .}. The relator a2 = e essential means to replace
a2 with e in all word in F1 . It should be clear that the only elements left after the substitutions are e and
a. Hence this group is isomorphic to the only group with two elements, Z2 .
2. Graphs
A little more graph theory is needed.
Definition 10. 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 (u0 , v 0 ) if and only if either
(1) u = u0 and vv 0 ∈ E(H), or
(2) v = v 0 and uu0 ∈ E(G).
Example 16. Figure 3 shows an example of how to take the Cartesian product of two graphs.
Definition 11. 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.
Example 17. Figure 4 shows examples of directed graphs, along with their Cartesian product.
CAYLEY GRAPHS
a
1
2
5
c
b
(2, a)
(1, a)
(1, b)
(1, c)
(2, b)
(2, c)
Figure 4. Cartesian product of directed graphs.
3. Cayley Graphs
Now that group basics and some additional graph theory have been given, Cayley graphs may be discussed.
Definition 12. 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 .
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.
Example 18. The Cayley graph of a group depends on the group, as well as the presentation of the group;
this is shown by Figures 3, 6, and 7, which are Cayley graphs of different presentations for the group Z.
a−2
a−1
a0
a1
a2
Figure 5. Cayley graph of ha | i, this is the canonical Cayley graph of Z.
a−1
a−2
a1
a0
a3
a2
a5
a4
a6
Figure 6. Cayley graph of ha, a2 | i.
a−4
a−3
a−2
a−1
a0
a1
a2
a3
Figure 7. Cayley graph of ha2 , a3 | i.
a4
6
RYAN JENSEN
Example 19. Figure 8 show the canonical Cayley graph of F2 .
Figure 8. Cayley graph of F2 = ha, b | i, this is the canonical Cayley graph of F2 .
Example 20. Figures 9, 10, and 11 show some Cayley graphs of common groups.
a2
a0
a1
Figure 9. Cayley graph of Z3 = ha | a3 i.
a1
a2
a0
a3
a4
Figure 10. Cayley graph of Z5 = ha | a5 i.
CAYLEY GRAPHS
7
Figure 11. Cayley graph of Z ⊕ Z = ha, b | aba−1 b−1 i.
4. Cayley Color Graphs
Very similar to a Cayley graph is a Cayley color graph. The only difference is that in the color graph,
inverses are not included by default. This makes it necessary to look at directed graphs instead of simple
graphs.
Definition 13. 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.
4.1. Theorems. Some Theorems about Cayley color graphs are now given, although their proof generally
are not. Some proofs are asked for as homework questions.
Theorem 2. The degree of each vertex in a Cayley color graph is equal to the total number of generators.
Theorem 3. Relators in a group presentation correspond to cycles in the Cayley color graph.
Theorem 4. A group is abelian if and only if for each pair of generators a, b, the path aba−1 b−1 is closed.
Theorem 5. Every finite group has a finite presentation, and hence has a Cayley color graph.
Theorem 6. 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.
Corollary 7. If ∆C (hS1 | R1 i) ∼
= ∆C (hS2 | R2 i), then hS1 | R1 i ∼
= hS2 | R2 i.
Theorem 8. 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 (hs1 , . . . , sm | r1 , . . . rt i) ∆C (hsm+1 , . . . , sn | rt+1 , . . . rq i)
−1
= ∆C (hs1 , . . . sn | r1 , . . . , rq , si sj s−1
i sj i)
for all 1 ≤ i ≤ m ≤ j ≤ n.
Proofs of the Theorems in this section are either trivial, or can be found in [10].
4.2. Examples. The follow Figures show more examples of Cayley color graphs.
8
RYAN JENSEN
a
e
b2
b
ab2
ab
Figure 12. ∆C (D3 ), where D3 = ha, b | a2 , abab, b3 i.
(12)
()
(123)
(132)
(23)
(13)
Figure 13. ∆C (S3 )
(1, 0)
(0, 0)
(0, 1)
(0, 2)
(1, 1)
(1, 2)
Figure 14. ∆C (Z2 ⊕ Z3 )
5. Home Work
You may pick and choose which questions/parts you want to do. Your grade will be out of 50 points,
with 50 being the maximum.
Question 1. The standard presentation for Dn is
Dn = ha, b | an , abab, b2 i,
the Cayley color graph for D3 was shown in Figure 12.
The definition of Dn can be extended to an infinite dihedral group, defined by
D∞ = ha, b | abab, b2 i.
CAYLEY GRAPHS
(a)
(b)
(c)
(d)
9
Draw the Cayley color graph for D4 . (10 points)
Describe the Cayley color graph for Dn . (10 points)
Draw the Cayley color graph for D∞ . (10 points)
Is D5 abelian, use graph theory to explain. (10 points)
Question 2.
(a) Draw the Cayley color graph for ha2 , a4 | i and compare it to the graph in Figure 7. (5 points)
(b) Prove or disprove: a ∈ han , am | i if and only if gcd(n, m) = 1. (10 points)
Question 3.
(a) Draw the Cayley color graph of Z2 ⊕ Z2 . (5 points)
(b) Draw the Cayley color graph of Z2 ⊕ Z4 . (5 points)
(c) Draw the Cayley color graph of Z2 ⊕ Z. (5 points)
(d) Draw the Cayley color graph of Z2 ⊕ Z2 ⊕ Z2 . (10 points)
(e) From the Cayley color graph of Z2 ⊕ Z2 ⊕ Z2 , give a group presentation of Z2 ⊕ Z2 ⊕ Z2 . (15 points)
Question 4.
(a) Prove Theorem 4. (5 points)
(b) Prove Theorem 5. (5 points)
Question 5.
(a) Using the Cayley color graph of S3 shown in 13, find a subgroup of S3 . (10 points)
(b) How many subgroups of S3 are there? (10 points)
(c) Using the Cayley color graph of D4 , find a subgroup of D4 (see Question 1.a). (10 points)
(d) How many subgroups of D4 are there?
Question 6. A subgroup N of a group G is normal if gng −1 ∈ N for all g ∈ G and n ∈ N .
(a) Find a normal subgroup of S3 . (15 points)
(b) Show that any subgroup of Z is normal. (15 points)
Question 7.
(a) Draw the Cayley color graph for A4 = ha, b | a2 , b3 , (ab)3 i. (30 points)
(b) Show that Z3 is not normal in A4 . (15 points)
References
[1] Brian H. Bowditch, A course on geometric group theory, MSJ Memoirs, vol. 16, Mathematical Society of Japan, Tokyo,
2006. MR 2243589 (2007e:20085)
[2] 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.
[3] David S Dummit and Richard M Foote, Abstract algebra, (2004), John Wiley and Sons, Inc.
[4] Thomas W Hungerford, Algebra, volume 73 of graduate texts in mathematics, Springer-Verlag, New York, 1980.
[5] Bernard Knueven, Graph automorphisms, 2014.
[6] Serge Lang, Algebra revised third edition, Springer-Verlag, 2002.
[7] James Munkres, Topology (2nd edition), 2 ed., Pearson, 2000.
[8] Piotr W Nowak and Guoliang Yu, Large scale geometry, 2012.
[9] Douglas B. West, Introduction to graph theory (2nd edition), 2 ed., Pearson, 2000.
[10] A.T. White, Graphs of groups on surfaces, volume 188: Interactions and models (north-holland mathematics studies), 1
ed., North Holland, 5 2001.
Department of Mathematics, University of Tennessee, Knoxville, TN 37996
E-mail address: jensen@math.utk.edu