Cayley Graphs Ryan Jensen Groups Group Basics Examples Cayley Graphs Isomorphisms Forming Groups Free Groups Examples Relators Ryan Jensen Graphs Cayley Graphs University of Tennessee F 1 F 2 Presentations March 26, 2014 Cayley Color Graphs Examples Applications References
Group Cayley Graphs Definition Ryan Jensen A group is a nonempty set G with a binary operation ∗ which Groups satisfies the following: Group Basics Examples (i) closure: if a , b ∈ G , then a ∗ b ∈ G . Isomorphisms Forming Groups (ii) associative: a ∗ ( b ∗ c ) = ( a ∗ b ) ∗ c for all a , b , c ∈ G . Free Groups Examples (iii) identity: there is an identity element e ∈ G so that Relators Graphs a ∗ e = e ∗ a = a for all a ∈ G . Cayley Graphs (iv) inverse: for each a ∈ G , there is an inverse element F 1 F 2 a − 1 ∈ G so that a − 1 ∗ a = a ∗ a − 1 = e . Presentations Cayley Color A group is abelian (or commutative) if a ∗ b = b ∗ a for all Graphs Examples a , b ∈ G . Applications References 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 Example: Z Groups The integers Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } form an abelian Group Basics group under the addition operation. Examples Isomorphisms Forming Groups Free Groups Example: Z 2 Examples Relators Define Z / 2 Z = Z 2 = { ¯ 0 , ¯ 1 } , where ¯ 0 = { z ∈ Z | z is even } , and Graphs ¯ 1 = { z ∈ Z | z is odd } . Then Z / 2 Z is an abelian group. Cayley Graphs F 1 F 2 Presentations Example: Z n Cayley Color Graphs Let n ∈ Z , and define Z / n Z = Z n = { ¯ 0 , ¯ 1 , . . . n − 1 } , where Examples ¯ i = { z ∈ Z | remainder of z | n = i } are known as the integers Applications modulo n . Then Z / n Z is an abelian group. References
A closer look at Z 5 Cayley Graphs Ryan Jensen A multiplication (addition) table is called a Cayley Table . Let’s look at the Cayley table for the group Z 5 = { 0 , 1 , 2 , 3 , 4 } . Groups Group Basics Examples ∗ 0 1 2 3 4 Isomorphisms Forming Groups 0 0 1 2 3 4 Free Groups 1 1 2 3 4 0 Examples Relators 2 2 3 4 0 1 Graphs 3 3 4 0 1 2 Cayley Graphs F 1 4 4 0 1 2 3 F 2 Presentations Notice the table is symmetric about the diagonal, meaning the Cayley Color Graphs group is abelian. Examples Also 1 generates the group, meaning that if we add 1 to itself Applications enough times, we get the whole group. References
Other Examples of Groups Cayley Graphs There are many examples of groups, here are a few more: Ryan Jensen Examples of Groups Groups Group Basics Examples Isomorphisms GL ( n , R ), the general linear group over the real numbers , is Forming Groups the group of all n × n invertible matrices with entries in R . Free Groups Examples SL ( n , R ), the special linear group over the real numbers , is Relators Graphs the group of all n × n invertible matrices with entries in R Cayley Graphs whose determinant is 1. F 1 F 2 GL (2 , Z 13 ) is the group of 2 × 2 invertible matrices with Presentations Cayley Color entries from Z 13 (as before Z 13 is a group; it is actually a Graphs field since 13 is prime, but this won’t actually be needed in Examples Applications this presentation). References
Other Examples of Groups Cayley Graphs Ryan Jensen Groups Group Basics Examples of Groups Examples Isomorphisms Forming Groups S n , the symmetric group on n elements , is the group of Free Groups Examples bijections between an n element set and itself. Relators Graphs D n , the dihedral group of order 2 n , is the group of Cayley Graphs symmetries of a regular n -gon. F 1 F 2 Many others. Presentations Cayley Color Graphs Examples Applications References
Group Isomorphisms Cayley Graphs Ryan Jensen Definition Groups Group Basics Examples Let H and G be groups. A function f : G → H so that Isomorphisms Forming Groups f ( ab ) = f ( a ) f ( b ) for all a , b ∈ G is a homomorphism . Free Groups If f is bijective, then f is an isomorphism . Examples Relators If G = H , then f is an automorphism . Graphs Cayley Graphs If there is an isomorphism between G and G , then G and F 1 H are isomorphic , written G ∼ = H . F 2 Presentations Cayley Color Graphs Group isomorphisms are nice since they mean two groups are Examples the same except for the labeling of their elements. Applications References
Subgroups Cayley Graphs Ryan Jensen Definition Groups A subset H of a group G is a subgroup if is itself a group under Group Basics Examples the operation of G ; that H is a subgroup of G is denoted Isomorphisms Forming Groups H ≤ G . Free Groups Examples Relators Definition Graphs Cayley Graphs If Y is a subset of a group G , then the subset generated by Y F 1 is the collection of all (finite) products of elements of Y . This F 2 Presentations subgroup is denoted by � Y � . If Y is a finite set with elements Cayley Color Graphs y 1 , y 2 , . . . y n , then the notation � y 1 , y 2 , . . . y n � is used. A group Examples which is generated by a single element is called cyclic . Applications References
Examples of Subgroups Cayley Graphs Ryan Jensen Example: Trivial Subgroups Groups For any group G , the group consisting of only the identity is a Group Basics Examples subgroup of G , and G is a subgroup of itself. Isomorphisms Forming Groups Free Groups Example: Even Odd Integers Examples Relators Graphs A somewhat less trivial example is that the even integers are a Cayley Graphs subgroup of Z ; however, the odd integers are not as there is no F 1 identity element. F 2 Presentations Cayley Color Graphs Example: n Z Examples Applications For any integer n ∈ Z , n Z = { nz | z ∈ Z } is a subgroup of Z . References
Cartesian Product Cayley Graphs Ryan Jensen Definition Groups Group Basics Let A and B be sets. The Cartesian product of A and B is the Examples Isomorphisms set Forming Groups A × B = { ( a , b ) | a ∈ A , b ∈ B } Free Groups Examples Relators Graphs Example Cayley Graphs F 1 Let A = { 1 , 2 } and B = { a , b , c } then F 2 Presentations Cayley Color A × B = { (1 , a ) , (1 , b ) , (1 , c ) , (2 , a ) , (2 , b ) , (2 , c ) } Graphs Examples Applications References
Direct Product Cayley Graphs Ryan Jensen Groups Definition Group Basics Examples Given two groups G and H , their Cartesian product G × H , Isomorphisms Forming Groups (denoted G ⊕ H if G and H are abelian) is a group known as Free Groups the direct product ( direct sum if G and H are abelian) of G Examples Relators and H . The group operation on G × H is done coordinate-wise. Graphs Cayley Graphs Example: Z 2 ⊕ Z 3 F 1 F 2 Presentations There is a group of order 6 found by taking the direct sum of Cayley Color Graphs Z 2 and Z 3 , G = Z 2 ⊕ Z 3 . Examples Applications References
Quotient Groups Cayley Graphs Ryan Jensen Without going into too many technicalities about cosets, Groups normal subgroups etc., quotient groups can be defined. Group Basics Examples Isomorphisms Definition Forming Groups Free Groups Let G be a group and H a normal subgroup of G . Then the Examples quotient G / H is called the quotient group of G by H , or simply Relators Graphs G mod H . Cayley Graphs F 1 Example: Z / n Z F 2 Presentations Cayley Color Z is a group, and n Z is a normal subgroup of Z . So the Graphs Examples quotient Z / n Z is a group. (Remember Z / n Z = Z n .) Applications References
Free Groups Cayley Graphs Definition Ryan Jensen Let A be a set. Groups Group Basics The set A = { a 1 , a 2 , . . . } together with its formal inverses Examples Isomorphisms A − 1 = { a − 1 1 , a − 1 2 , . . . } from an alphabet . Forming Groups The elements of A ∪ A − 1 are called letters . Free Groups Examples Relators A word is a concatenation of letters. Graphs A reduced word is a word where no letter is adjacent to its Cayley Graphs F 1 inverse. F 2 Presentations The collection of all finite reduce words on the alphabet A Cayley Color Graphs is a free group on A , denote by F ( A ). Examples The group operation is concatenation of words, followed Applications References by reduction if necessary.
More Notation Cayley Graphs Ryan Jensen Groups Theorem Group Basics Examples Isomorphisms Let A and B be finite sets, then F ( A ) is isomorphic to F ( B ) if Forming Groups and only if | A | = | B | . Free Groups Examples Relators The above Theorem says that only the size of the alphabet is Graphs important when constructing a free group. As a result, when Cayley Graphs F 1 the alphabet is finite, i.e. | A | = n , the free group on A is F 2 Presentations denoted F n and is called the free group of rank n , or the free Cayley Color group on n generators . Graphs Examples Applications References
Examples of Free Groups Cayley Graphs Example: Trivial Free Group Ryan Jensen The free group on an empty generating set (or the free group Groups Group Basics on 0 generators) is the trivial group consisting of only the Examples Isomorphisms empty word (the identity element). Forming Groups Free Groups Examples F ( ∅ ) = F 0 = { e } . Relators Graphs Cayley Graphs F 1 Example: F 1 F 2 Presentations The free group on one generator is isomorphic to the integers. Cayley Color Graphs F 1 = { . . . , a − 2 , a − 1 , a 0 = e , a = a 1 , a 2 , . . . } Examples Applications = Z by the map a i �→ i . F 1 ∼ References
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