Groups

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# Groups

Mainly to fix notations and for the sake of completeness we recall here some well know definitions and constructions that we will need.

Consider a group , a subset of is called a set of generators iff the smallest subgroup that contains is all . Given a set of generators a finite sequence of the symbols is called a word on the generators; we say that the generators satisfy the relation , where is a word on them, if the element obtained multiplying the symbols of is the identity; a word is said to be in reduced form if all the pairs of the form or have been removed from it, as this could left an empty word we use the convention that the empty word is in reduced form and call it the identity word, we usually represent it by .

We would like to describe groups just naming their generators and relations, but often, if not always this is just impossible, for instance the Klein group is generated by two elements such that , but also the group of two elements satisfies these properties with . One could argue that what we want is the largest group that satisfy those relations, but then the problem of the existence of such a group arise. Fortunately one can prove that there is always such a group, we reproduce here the construction because it is interesting in what follows.

We first build a group that contains the generators . Consider the set of reduced words on the generators with the operation of composition, it is easy to check that this is a group, where the identity element is the empty word (thus the name we gave it above), and the inverse of a word is the word . Notice that in fact the generators ``belong'' to this group, each represented by the sequence . The group just built is called the free group on the generators, we will write it as or simply .

The given relations are members of , so we can consider the smallest normal subgroup of that contains all the relations, and let us call the quotient . Now does satisfy the prescribed relations, to prove that in the largest possible group let be a group generated by a set that satisfy the relations, then there is a natural function

that maps each word to the product of the symbols in , it is easy to see that this map is an homomorphism of groups, and that its kernel contains , because is a normal subgroup, and since satisfy the relations they are mapped to the identity in , so , from the isomorphism theorems follow that is an extension of , so is the largest group generated by subject to the given restrictions.

A group is said to have presentation

if is isomorphic to the largest group generated by the elements that satisfies the relations . Of course the same group can have many presentations.

We now prove that it is possible to find the multiplication table of a finite group given a presentation for it, it is known that this is not possible for infinite groups, but we do not address that problem here.

Consider a group (possibly infinite) with a presentation

were is a word in the generators , we can associate a graph to the pair as follows, the vertices of the graph are the elements of , we join two vertices and by an edge if there is a generator such that , we say that this edge has color , notice that the edge is naturally oriented leaving and arriving to , we add another oriented edge from to with color . This graph is called the Cayley graph of for the presentation , notice that in fact the graph depends on the presentation, to see this is enough to add a new generator to a given presentation, then the new graph contains two more edges on each vertex.

Given a word on the generators and a vertex in there is only one path such that it starts on and its edges are colored , conversely to each path we can associate a word which we will call the name of the path. Notice that if a path starts in the element and has name its ending vertex is exactly the one associated to the element in which the word represents.

The Cayley graph can be used to find the multiplication table of , since the set generates for each element in there is a word in the generators such that , this means that the path starting from with name ends on the vertex , notice also that the path starting from with name ends on the element , thus we can find the product of and simply finding the ending vertex of the path . Even more the words induce a labeling in the nodes, we can use that labeling to recognize the elements and write the multiplication table.

We also just proved that the Cayley graph is connected, because each element is connected to the vertex by the path with name , it can be easily proved too that a path is a loop iff its name represents the identity in .

In this way our original problem has been reduced to finding the Cayley graph of the group, although at first this doesn't look very promising (since we don't even have the order of the group, that is we don't even know how many vertices the graph has) we will prove that the Cayley graph really solves our problem.

We first observe that a Cayley graph satisfies the following properties

1.   It is not empty and connected.

The first assertion comes from the fact that is non empty, above we proved the second.

2.   On each vertex incides exactly one edge for each color.

3.   If from vertex originates an edge of color arriving to vertex then from originates an edge of color to .

This last two properties come from the definition of

4.   To each word in the generators of we can associate a unique path starting from the node .

This comes from the previous properties.

5.   A path is a loop iff its name belong to (the normal subgroup defined above).

We noticed that a path is a loop iff its name is the identity in , but then its name must belong to from the definition of .

Suppose we have colored graph with vertex set and colors that satisfies the properties above, we can define a function

that maps each word to the ending vertex of the path defined in property . Consider the equivalence relation on defined by

is clear that there is a bijection between and . On the other hand if then is a loop, but then , so , where is the natural map

so is exactly the equivalence relation in the quotient , then there is a and onto map

We want to extend to a isomorphism of graphs between and , in fact, two vertices are joined by an edge of color iff the ending vertex of is the vertex iff iff iff iff is joined to by an edge of color .

Figure: An algorithm to calculate

Let us now consider the algorithm in figure , a vertex is called ``filled'' if has all the needed edges, i.e. there is at least one edge of each color on it, the algorithm ``fills'' a vertex adding all the needed edges, since the destination of the edges are unknown at first an extra vertex is added as destination, is the starting vertex of the path and is the ending vertex of it, step (2.2.1) makes the path a loop, this step may ``overfill'' the new (identified) vertex because two or more edges of the same color may contain it, this induces new identifications on the neighbors of the vertex (to eliminate the extra edges), this identifications must be propagated along the graph.

It is easy to see that if the algorithm stops then the graph obtained satisfy the properties above, so it is isomorphic to .

We will now prove that that if is finite the algorithm stops, although this is quite intuitive a formal proof gives some insight.

Let be the graph produced after iterations before entering the external loop (step (2)), notice that in the path is well defined, if , where is the length of the word ; for each let be the shortest word (lexicographically minimal if there are several) that satisfy , then after at most iterations the graph produced by the algorithm contains at least one vertex for each element , namely .

For any and define , we want to prove that after a finite number of iterations the vertices and are joined by an edge of color , we now that after at most iteration has all its edges in particular of color , if the destination of is then there is nothing to prove, otherwise let be the destination of , then , but then there is a word such that , but

where y , but then after at most

the step (2.2) of the algorithm warrants that has been identified with . Since there is a finite number of pairs after a finite number of iterations () the graph contains an isomorphic copy of , namely the subgraph generated by the vertices where , but then those are all the vertices of , if not so because the graph is connected, there must be one vertex from where an edge originates to an extra vertex , but from the construction there is only one edge for each color coming out of and all of them go to the vertices, this is a contradiction. At last notice that all the vertices are filled, so the algorithm stops.

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coryan@mat.puc.cl