Recently, we've been looking at group presentations, where a group is presented as a set of generators together with a set of relations that hold between those generators. Group elements are then represented as words in the generators.
One can then ask questions about these words, such as: What is the longest (reduced) word in the group? How many (reduced) words are there of each length?
This week I want to look at Cayley graphs, which are a way of visualising groups. Questions about word length translate to questions about path distance in the Cayley graph.
So, the Cayley graph of a group, relative to a generating set gs, is the graph
- with a vertex for each element of the group
- with an edge from x to y just whenever x*g = y for some generator g in gs
Notice that as we have defined it, the edges are directed (from x to y), so this is a directed graph, or digraph.
Here's the Haskell code:
module Math.Algebra.Group.CayleyGraph where
import Math.Algebra.Group.PermutationGroup as P
import Math.Algebra.Group.StringRewriting as SR
import Math.Combinatorics.Graph
import qualified Data.List as L
import qualified Data.Set as S
data Digraph a = DG [a] [(a,a)] deriving (Eq,Ord,Show)
-- Cayley digraph given a group presentation of generators and relations
cayleyDigraphS (gs,rs) = DG vs es where
rs' = knuthBendix rs
vs = L.sort $ nfs (gs,rs') -- reduced words
es = [(v,v') | v <- vs, v' <- nbrs v ]
nbrs v = L.sort [rewrite rs' (v ++ [g]) | g <- gs]
-- Cayley digraph given group generators as permutations
cayleyDigraphP gs = DG vs es where
vs = P.elts gs
es = [(v,v') | v <- vs, v' <- nbrs v ]
nbrs v = L.sort [v * g | g <- gs]
As an example, let's look at the Cayley digraph of the dihedral group D8 (the symmetries of a square), generated by a rotation and a reflection:
> :load Math.Algebra.Group.CayleyGraph
> let a = p [[1,2,3,4]]
> let b = p [[1,2],[3,4]]
> a^3*b == b*a
True
> cayleyDigraphS (['a','b'],[("aaaa",""),("bb",""),("aaab","ba")])
DG ["","a","aa","aaa","aab","ab","b","ba"] [("","a"),("","b"),("a","aa"),("a","ab"),("aa","aaa"),("aa","aab"),("aaa",""),("aaa","ba"),("aab","aa"),("aab","ab"),("ab","a"),("ab","b"),("b",""),("b","ba"),("ba","aaa"),("ba","aab")]
> cayleyDigraphP [a,b]
DG [[],[[1,2],[3,4]],[[1,2,3,4]],[[1,3],[2,4]],[[1,3]],[[1,4,3,2]],[[1,4],[2,3]],[[2,4]]] [([],[[1,2],[3,4]]),([],[[1,2,3,4]]),([[1,2],[3,4]],[]),([[1,2],[3,4]],[[1,3]]),([[1,2,3,4]],[[1,3],[2,4]]),([[1,2,3,4]],[[2,4]]),([[1,3],[2,4]],[[1,4,3,2]]),([[1,3],[2,4]],[[1,4],[2,3]]),([[1,3]],[[1,4,3,2]]),([[1,3]],[[1,4],[2,3]]),([[1,4,3,2]],[]),([[1,4,3,2]],[[1,3]]),([[1,4],[2,3]],[[1,3],[2,4]]),([[1,4],[2,3]],[[2,4]]),([[2,4]],[[1,2],[3,4]]),([[2,4]],[[1,2,3,4]])]These are of course the same Cayley digraph, just with different vertex labels. Here's a picture:
The picture might remind you of something. You can think of a Cayley digraph as a state transition diagram, where the states are the group elements, and the transitions are multiplication (on the right) by g, for each generator g. It might help to think of each edge as being labelled by the generator that caused it.
A few things to notice.
First, Cayley digraphs are always regular: the out-degree of each vertex, the number of edges leading out of it, will always equal the number of generators; and similarly for the in-degree, the number of edges leading into each vertex. (Exercise: Prove this.) In fact, we can say more - the graph "looks the same" from any vertex - this follows from the group properties. (Exercise: Explain.)
Second, notice how some of the edges come in pairs going in opposite directions. Why is that? In this case, it's because one of our generators is its own inverse (which one?) - so if it can take you from x to y, then it can take you back again. In general, whenever our set of generators contains a g such that g^-1 is also in the set, then the edges corresponding to g, g^-1 will come in opposing pairs.
Given this, we can omit the arrows on the edges if we adopt the convention that whenever we are given a set of generators, their inverses are also implied. In this way, we obtain an undirected or simple graph. Here's the code:
toSet = S.toList . S.fromList
-- The Cayley graph on the generators gs *and their inverses*, given relations rs
cayleyGraphS (gs,rs) = graph (vs,es) where
rs' = knuthBendix rs
vs = L.sort $ nfs (gs,rs') -- all reduced words
es = toSet [ L.sort [v,v'] | v <- vs, v' <- nbrs v ] -- toSet orders and removes duplicates
nbrs v = [rewrite rs' (v ++ [g]) | g <- gs]
cayleyGraphP gs = graph (vs,es) where
vs = P.elts gs
es = toSet [ L.sort [v,v'] | v <- vs, v' <- nbrs v ]
nbrs v = [v * g | g <- gs]
For example:
> cayleyGraphS (['a','b'],[("aaaa",""),("bb",""),("aaab","ba")])
G ["","a","aa","aaa","aab","ab","b","ba"] [["","a"],["","aaa"],["","b"],["a","aa"],["a","ab"],["aa","aaa"],["aa","aab"],["aaa","ba"],["aab","ab"],["aab","ba"],["ab","b"],["b","ba"]]
One important point to note is that the Cayley graph of a group is relative to the generators. For example, we saw last time that the dihedral groups can also be generated by two reflections. In the case of D8, we can set r = (1 2)(3 4), s = (1 3).
> let r = p [[1,2],[3,4]]
> let s = p [[1,3]]
> (r*s)^4
[]
> cayleyGraphS (['r','s'],[("rr",""),("ss",""),("rsrsrsrs","")])
G ["","r","rs","rsr","rsrs","s","sr","srs"] [["","r"],["","s"],["r","rs"],["rs","rsr"],["rsr","rsrs"],["rsrs","srs"],["s","sr"],["sr","srs"]]
So the point to emphasize is that this graph and the one shown previously are Cayley graphs of the same group. The vertices represent the same elements (considered as permutations). However, because we have taken different sets of generators, we get different edges, and hence different graphs.
Ok, so what does the Cayley graph tell us about the group? Well, as an example, consider the Cayley graph of the Rubik cube group, as generated by the face rotations f, b, l, r, u, d (as defined previously). The vertices of the graph can be identified with the possible positions or states of the cube. The group element 1 corresponds to the solved cube. The edges correspond to single moves that can be made on the cube. If someone gives you a scrambled cube to solve, they are asking you to find a path from that vertex of the Cayley graph back to 1.
Given any graph, and vertices x and y, the distance from x to y is defined as the length of the shortest path from x to y. On the Rubik graph (ie, the Cayley graph of the Rubik cube), the distance from x to 1 is the minimum number of moves needed to solve position x. The HaskellForMaths library provides a distance function on graphs. Thus for example:
> let graphD8rs = cayleyGraphS (['r','s'],[("rr",""),("ss",""),("rsrsrsrs","")])
> distance graphD8rs "" "rsr"
3The distance from 1 to an element g is of course the length of the reduced word for g.
The diameter of a graph is defined as the maximum distance between vertices. So the diameter of the Rubik graph is the maximum number of moves that are required to solve a scrambled position. It has recently been shown that this number is twenty.
> diameter graphD8rs
4The diameter of a Cayley graph is the length of the longest reduced word.
That's really all I wanted to say for the moment. Next time, we'll take some of these ideas further.
