This page lists various constructions of new examples of groups and subgroups from known examples, and gives some brief notes of what each construction is good for.
This is a list of various standard constructions of groups and subgroups.
Amalgamated free products
Graphs of groups
The group law on the direct product is determined by the requirement that for every and . That is, acts on by conjugation, and the action is trivial. The semidirect product construction derives from the observation that any action of on could be used to construct a new group in the same way. Let be a right-action of on , so for any we have an automorphism of that acts on the right, sending an element to . The semidirect product is in bijection with the set of pairs where and , and the group law is determined by the requirement that .
Often the action is suppressed. By construction, the subgroup of is normal, and the quotient is isomorphic to . So is an extension of by . In fact, any split extension of by is a semidirect product.
Two geometric examples of semidirect products are the dihedral group , which is and the Euclidean group, which is . For the dihedral group, the normal subgroup can be interpreted as the group of rotations by and the factor is some reflection. The action of reflections on the rotations simply reverses the direction of rotation. To see this, note that rotating by then reflecting is the same as reflecting and then rotating by .
For the Euclidean group, the normal subgroup is -dimensional translations, and the action is the same as rotating the coordinates of the translation.
A group is an extension of a group by a group if there is a short exact sequence
(In other words, the is the quotient of by .) The following example takes advantage of two features of extensions.
The subgroup is to as the trivial subgroup is to .
Often can be chosen to have better properties than . In particular, any presentation for corresponds to an extension where is free.
It is a famous and non-trivial fact that there exists a finite group presentation
in which the word problem is unsolvable. That is, there is no algorithm that will tell you whether or not a given word in the generators represents the identity in . (We will not use the fact that is finite in this example. But it will be important in Example 3 below.)
We will use group extensions to produce a different pathology in a much better behaved group—a free group.
Let be the free group on . The relations can be thought of as elements of . By the universal property of free groups, the obvious map extends to a surjection . The kernel of this surjection is precisely , the normal subgroup of generated by the relations. That is, we have a short exact sequence
The fact that the word problem is unsolvable in can now be restated precisely as the assertion that there are normal subgroups of with unsolvable membership problem.
So we see that the existence of the highly pathological group corresponds to a different sort of pathological behaviour in the well-behaved group
The fibred product construction in the category of groups is the same as in the category of sets. If and are surjections then the fibred product of and is the subgroup of defined as the preimage of the diagonal subgroup of under the map . That is,
Fibred products can be used to improve the finiteness properties of subgroups. Example 2 showed how to construct a non-trivial subgroup of a free group with unsolvable membership problem. Although was finitely generated as a normal subgroup of , it is a consequence of Greenberg's Theorem (Greenberg's Theorem states that every finitely generated normal subgroup of a finitely generated free group is of finite index) that is not finitely generated as a group. In fact, every finitely generated subgroup of a free group has solvable membership problem.
In the following example, we will use a fibred product to construct a finitely generated subgroup of a direct product of two free groups that has unsolvable membership problem.
Let be a finitely presented group with unsolvable word problem as in Example 2 and let be the quotient map derived from the presentation. Let be the fibre product of two copies of , a subgroup of . Then
where is the diagonal subgroup of . The membership problem for in is unsolvable, (indeed, is an element of if and only if is trivial in ) and this translates precisely to the statement that the membership problem for in is unsolvable. But the finite set
We have proved the following.
A wreath product is a special case of a semidirect product. We will first restrict our attention to the wreath product of two finite groups and . The set of set maps
is a group (multiplication comes from multiplication in ) and is naturally equipped with a right-action of , namely the action by left translation. (It is easy to get confused by the fact that left translation is a right action!) We can think of as the direct sum of copies of , indexed by the elements of , and acts by permuting the factors. This is precisely the data needed for a semidirect product construction.
The wreath product of by is defined to be
where acts on by left translation.
When or may be infinite, we define to be precisely those set maps that equal the identity on all but finitely many elements of . This has the effect that is isomorphic to the direct sum, rather than the direct product, of copies of . Our first example of a wreath product shows how to construct a 2-generator group with an abelian subgroup of infinite rank.
The group is the direct sum of countably many copies of , and so can be thought of as the group of biinfinite sequences of integers that are equal to zero in all but finitely many coordinates. It admits an action of , where the integer acts by moving the th coordinate to the th coordinate. The orbit of the sequence that is in every non-zero coordinate and in the th coordinate generates . (Here the fact that is the direct sum, rather than the direct product, is important.) The resulting semidirect product is precisely the wreath product . It contains the infinite-rank abelian group as a subgroup, and is generated by just two elements.
More generally, given any transitive action of on a set , one can define the wreath product to be the semidirect product of and , where as before acts on by left-translation.
Amalgamated free products
Amalgamated free products are push-outs in the category of groups. If and are both injective then the amalgamated free product is defined by the property that the diagram
is a push out. That is, is the freest possible group that contains and as subgroups in which the two copies of are identified.
The Seifert–van Kampen Theorem asserts that if a path-connected topological space can be decomposed as the union of two closed, path-connected subsets whose intersection is also path-connected then the fundamental group of is a push-out. Therefore, amalgamated free products arise very naturally in topology.
Suppose is a compact orientable surface and is a simple closed curve that is not homotopic to a point. Suppose further that is separating—that is, has two path-components; we shall denote their closures by and . Because is not homotopic to a point, the natural inclusions are injective at the level of for . (This follows from the classification of surfaces.) Therefore
by the Seifert–van Kampen Theorem.
One of the most important results for proving theorems about amalgamated free products is the Normal Form Theorem, which gives a criterion that determines when elements of , expressed as products of elements of the images of and , are nontrivial. It is a consequence that the maps and are both injective. For more details, see the article How to prove facts about graphs of groups. We will describe the normal form theorem in the important special case of free products below.
If is the trivial group then the amalgamated product of and over is called the free product of and , denoted . This is the freest possible group that contains both and . The Normal Form Theorem for free products determines when elements of are non-trivial. Note that it is immediate from the definition that is generated by the union of and .
Example 5 explains what happen at the level of when you cut a surface along a separating curve. But the separating hypothesis is rather unnatural—it makes just as much sense to cut along a non-separating curve . What happens in this case? The answer is that decomposes as an HNN extension.
Suppose are both injective homomorphisms. If has presentation then the Higman–Neumann–Neumann (HNN) extension is
The generator is called the stable letter
Suppose is a compact orientable surface and is a simple closed curve that is not homotopic to a point. Suppose further that is non-separating, so has one path-component , and two-sided (that is, is not the core of a Möbius band). Then
by the Seifert–van Kampen Theorem.
There are two things to notice about this definition. The first is that, a priori, it seems to depend on the chosen presentation for —however, one can show that the definition is in fact independent of this choice. More importantly, is rather poor notation as it does not specify the maps and . Often, as in Example 6, the two maps are implicit. One way of getting round this ambiguity is to set and write instead of . Alternatively, we sometimes write .
The analogue of the Normal Form Theorem for amalgamated products in the context of HNN extensions is Britton's Lemma, which gives a criterion for when elements of an HNN extension are trivial. In particular, Britton's Lemma implies that the natural homomorphism is injective.
As is apparent from the definition, HNN extensions force elements to be conjugate. In a similar spirit to Example 4, one can use an HNN extension to reduce the number of generators required. However, HNN extensions are more flexible than semidirect products as they only require two isomorphic subgroups, rather than an automorphism of the whole group. The following example is related to the original application of HNN extensions by Higman, Neumann and Neumann.
We will give a proof of the following.
Let be any countable group. In order to be able to apply an HNN extension, we first need to modify so that it has two isomorphic subgroups. Consider , the free product of with an infinite cyclic group. For each , let and let . It follows from the normal form theorem for free products that the subgroup generated by is isomorphic to the free group on , for each . In particular, . So we can construct the HNN extension
If is the stable letter of the HNN extension then we have
for each , and so is generated by as required.
Graphs of groups
In Example 5 and Example 6, we saw what happens at the level of the fundamental group when an orientable surface is cut along an embedded curve. But what if we want to cut along several disjoint curves at once? (The union of finitely many disjoint curves is called a multicurve.)
Let be a compact, orientable surface and let be a finite set of disjoint simple closed curves. Assume that no is homotopic to a point. The complement of the images of the is a disjoint union of connected components :
Correspondingly, the fundamental group of decomposes as a graph of groups. The vertex groups are the fundamental groups of the , the edge groups are the cyclic subgroups generated by the , and the underlying graph has one vertex for each and one edge for each , with the obvious incidence relations.