Cayley Graphs. Ryan Jensen. March 26, University of Tennessee. Cayley Graphs. Ryan Jensen. Groups. Free Groups. Graphs.

Transcription

1 Cayley Forming Free Cayley Cayley University of Tennessee March 26, 2014

2 Group Cayley Forming Free Cayley 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. (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.

3 of Cayley Forming Free Cayley Example: Z The integers Z = {..., 2, 1, 0, 1, 2,...} form an abelian group under the addition operation. Example: Z 2 Define Z/2Z = Z 2 = { 0, 1}, where 0 = {z Z z is even}, and 1 = {z Z z is odd}. Then Z/2Z is an abelian group. Example: Z n Let n Z, and define Z/nZ = Z n = { 0, 1,... n 1}, where ī = {z Z remainder of z n = i} are known as the integers modulo n. Then Z/nZ is an abelian group.

4 A closer look at Z 5 Cayley Forming Free Cayley 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} 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.

5 Other of Cayley Forming Free Cayley There are many examples of groups, here are a few more: of 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 13 ) is the group of 2 2 invertible matrices with entries from Z 13 (as before Z 13 is a group; it is actually a field since 13 is prime, but this won t actually be needed in this presentation).

6 Other of Cayley Forming Free Cayley of S n, the symmetric group on n elements, is the group of bijections between an n element set and itself. D n, the dihedral group of order 2n, is the group of symmetries of a regular n-gon. Many others.

7 Group Cayley Forming Free Cayley 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. If G = H, then f is an automorphism. If there is an isomorphism between G and G, then G and H are isomorphic, written G = H. Group isomorphisms are nice since they mean two groups are the same except for the labeling of their elements.

8 Subgroups Cayley Forming Free Cayley 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. 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 Y. If Y is a finite set with elements y 1, y 2,... y n, then the notation y 1, y 2,... y n is used. A group which is generated by a single element is called cyclic.

9 of Subgroups Cayley Forming Free Cayley Example: Trivial Subgroups For any group G, the group consisting of only the identity is a subgroup of G, and G is a subgroup of itself. 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.

10 Cartesian Product Cayley Forming Free Cayley 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} 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)}

11 Direct Product Cayley Forming Free Cayley 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. Example: Z 2 Z 3 There is a group of order 6 found by taking the direct sum of Z 2 and Z 3, G = Z 2 Z 3.

12 Quotient Cayley Forming Free Cayley 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. 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 = Z n.)

13 Free Cayley Forming Free Cayley Definition Let A be a set. The set A = {a 1, a 2,...} together with its formal inverses A 1 = {a 1,...} from an alphabet. 1, a 1 2 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. 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.

14 More Notation Cayley Forming Free Cayley Theorem Let A and B be finite sets, then F (A) is isomorphic to F (B) if and only if A = B. 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 F n and is called the free group of rank n, or the free group on n generators.

15 of Free Cayley Forming Free 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). F ( ) = F 0 = {e}. Cayley Example: The free group on one generator is isomorphic to the integers. = {..., a 2, a 1, a 0 = e, a = a 1, a 2,...} = Z by the map a i i.

16 of Free Cayley Forming Free Cayley Example: 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, a 3, b 2, bab 1. An example of group operation: Example: F 3 (a 3 ) (b 2 ) (bab 1 ) = a 3 b 2 bab 1 = a 3 b 1 ab 1 F 3 on generators a, b, c is done in a similar manner.

17 Cayley Forming Free Cayley Definition Let F be a free group. 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. 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 S R where S is the set of generators and R the set of relators. S R is called a group presentation. For example {a, b} {aba 1 b 1 = e }, usually abbreviated a, b aba 1 b 1.

18 of Cayley Forming Free Cayley Example: Trivial is the trivial group consisting of only the empty word. a is. a, b is. Example: Z 2 The group given by the presentation a a 2 is isomorphic to Z 2. The letter a generates = {..., a 2, a 1, e, a, a 2...}. The relator a 2 = e means replace a 2 with e for all words in. The only elements left are e and a. Hence this group is isomorphic to Z 2.

19 A Little Graph Theory Cayley Forming Free Cayley 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 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.

20 Cartesian Product of Cayley 1 a Forming 2 b c Free (2, a) Cayley (1, a) (1, b) (1, c) (2, b) (2, c)

21 Directed Cayley Directed graphs just have directed edges. 1 a Forming 2 b c Free (2, a) Cayley (1, a) (1, b) (1, c) (2, b) (2, c)

22 Cayley Cayley Forming Free Cayley 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. 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.

23 Cayley Graph of Cayley Forming Free Cayley We will construct a Cayley Graph for = Z. Recall that = a = {... a 2, a 1, a 0, a 1, a 2,...}. We want to draw = (, a) = ( a ). The vertices of are the elements of. Take any element a i, since a is a generator, there is an edge between a i and a i a = a i+1. The result is an infinite graph, the real line R. a 2 a 1 a 0 a 1 a 2

24 Another Cayley Graph of Cayley Forming Free The Cayley Graph depends both on the group and on the generating set chosen. Lets look at ( a, a 2 ). a 1 a 1 a 3 a 5 Cayley a 2 a 0 a 2 a 4 a 6

25 Yet Another Cayley Graph of Cayley Forming Free Lets draw ( a 2, a 3 ). a 4 a 3 a 2 a 1 a 0 a 1 a 2 a 3 a 4 Cayley

26 Canonical Cayley Graph of Cayley Forming Free Lets look at ( a, b ). Cayley

27 Canonical Cayley Graph of Cayley Forming Free Cayley

28 of how to draw Cayley Cayley Forming Free Lets draw a a 2. a 4 a 3 a 2 a 1 a 0 a 1 a 2 a 3 a 4 Cayley a 0 a 1

29 of how to draw Cayley Cayley Forming Free Now lets do a a 3. a 4 a 3 a 2 a 1 a 0 a 1 a 2 a 3 a 4 Cayley a 2 a 0 a 1

30 of how to draw Cayley Cayley a a 5, in a different approach. Forming Free Cayley a 0 a 1 a 2 a 3 a 4 a 1 a 2 a 0 a 3 a 4

31 Cayley Forming Free Cayley From these examples, we can see that: a a n = {a 0, a 1,... a n 1 }, and generated by a. Z n = {0, 1,... n 1}, and generated by 1. So Z n = a a n by the map i a i. Z n is known as the cyclic group of order n, and the Cayley graph is the cyclic graph of length n.

32 of how to draw Cayley Cayley Forming Free ( a, b aba 1 b 1 ) aba 1 b 1 = e if and only if ab = ba. Cayley

33 Facts about Cayley Cayley Forming Free Cayley Facts The degree of each vertex is equal to the total number of generators, i.e. S S 1. 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. 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.

34 Cayley Forming Free Cayley A 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 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. So C ( a ) is a 2 a 1 a 0 a 1 a 2

35 of Cayley Forming Free C ( a a 5 ). a 2 a 1 Cayley a 0 a 3 a 4

36 of Cayley Forming Free Lets look at the Graph of a, b a 2, abab, b 3, which is a presentation for the group D 3. First notice that a 2 = 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. b 2 ab 2 Cayley b e ab a

37 of Cayley Forming Free We can redraw the graph. a e (2, a) (1, a) Cayley ab 2 b b 2 ab (1, b) (1, c) (2, b) (2, c) 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.

38 Theorems Cayley Forming Free Cayley Theorem Let C ( S R ) be a graph for a finite group. Then Aut ( S R ) = S R, this is not dependent on the presentation of the group. Corollary If C ( S 1 R 1 ) = C ( S 2 R 2 ), then S 1 R 1 = S 2 R 2.

39 Theorems Cayley Forming Free Cayley Theorem Let H and G be finite groups with presentations P H and P G. Then there is a presentation for H G so that Specifically, C (P H ) C (P G ) = C (P H P G ). C ( s 1,..., s m r 1,... r t ) C ( s m+1,..., s n r t+1,... r q ) = C ( s 1,... s n r 1,..., r q, s i s j s 1 i s 1 j ) 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.

40 of Cayley Forming Free Using the previous Theorem, we can find the standard Cayley color graph of Z 2 Z 3 (Z 2 = a a 2, and Z 3 = a a 3 ). C (Z 2 ) is a directed cycle of size 2 C (Z 3 ) is a directed cycle of size 3. Hence (standard) C (Z 2 Z 3 ) = C (Z 2 ) C (Z 3 ). (1, 0) Cayley (0, 0) (0, 1) (0, 2) (1, 1) (1, 2)

41 of Cayley Forming Free Cayley The Cayley of S 3 = a, b a 2, b 2, (ab) 3 is shown below. Note that S 3 is usually written S 3 = {(), (12), (13), (23), (123), (132)}, the vertices are labeled this way. (12) () (123) (132) (23) (13)

42 of Cayley Forming Free We can now analyze the groups D 3 and S 3. a e (12) () Cayley ab 2 b b 2 ab (123) (132) (23) (13) Since they Cayley color graphs are isomorphic, the groups D 3 and S 3 are isomorphic, even though they may not have the same presentation.

43 of Cayley Now lets look at the groups D 3 and Z 2 Z 3. Forming Free a e (1, 0) (0, 0) Cayley ab 2 b b 2 ab (0, 1) (0, 2) (1, 1) (1, 2) These groups are not isomorphic, as D 3 is not abelian, and Z 2 Z 3 is.

44 in Math Cayley Forming Free Cayley 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.

45 in Math Cayley Forming Free Cayley 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.

46 in Computer Science Cayley Forming Free Cayley 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.

47 in Computer Science Cayley Forming Free Cayley Here is an example group/cayley graph from their paper. Example from Langston et al. The group was a subgroup of GL(2, Z 13 ) consisting of all elements with determinant of ±1. The generators where [ ] [ ] order 2, order 52, [ ] order 14. The Cayley graph of this group has degree 5, diameter 7, and has 4368 vertices. A new record.

48 Cayley Forming Free Cayley Brian H. Bowditch, A course on geometric group theory, MSJ Memoirs, vol. 16, Mathematical Society of Japan, Tokyo, MR (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, 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, Bernard Knueven, Graph automorphisms, Serge Lang, Algebra revised third edition, Springer-Verlag, James Munkres, Topology (2nd edition), 2 ed., Pearson, Piotr W Nowak and Guoliang Yu, Large scale geometry, Douglas B. West, Introduction to graph theory (2nd edition), 2 ed., Pearson, A.T. White, of groups on surfaces, volume 188: Interactions and models (north-holland mathematics studies), 1 ed., North Holland,