Cayley Graphs

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