Quick description
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.
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General discussion
This is a list of various standard constructions of groups and subgroups.
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Direct products
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Semidirect products
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Extensions
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Central extensions
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Fibre products
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Wreath products
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Amalgamated free products
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Free products
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HNN extensions
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Graphs of groups

Direct products
Given groups and
, the simplest way we might think to construct a product group is by component-wise multiplication on the set
. This does give rise to a well-defined group operation, and the resulting group is called the direct product of
and
. We can naturally identify
with
and
with
, so we generally just consider
and
themselves to be subgroups of
.
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Semidirect products
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.
Example 1
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.
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Extensions
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.
Example 2
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
Fibred products
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.
Example 3
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

generates .
We have proved the following.




Wreath products
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.
Example 4
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.
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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
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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.
Example 5
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.
Free products
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
.
HNN extensions
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
Example 6
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.
Example 7
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.)
Example 8
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 the fundamental group of 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.
The definition of a graph of groups is rather technical, so we will not give it here. But the main ideas are captured by Example 8. Note that the multicurve might have happened to consist of just one connected component. Likewise, the definition of a graph of groups generalizes both the definitions of amalgamated products and HNN extensions. These correspond to the two cases in which the underlying graph has precisely one edge: the underlying graph of an amalgamated product is two vertices joined by an edge; the underlying graph of an HNN extension is a loop consisting of one vertex and one edge.
In Example 8, rather than cutting along the multicurve
we could have cut
along
, then cut the remaining pieces along
, and so on. This operation expresses
iteratively as a sequence of amalgamated products and HNN extensions. Likewise, any graph of groups can be thought of as built up from its vertex groups by taking iterative amalgamated free products and HNN extensions over its edge groups. So any group that you construct as a graph of groups could also have been constructed using amalgamated products and HNN extensions.
Graphs of groups really come into their own if you want to understand subgroups of amalgamated products and HNN extensions. To see why, let's return to the example of a curve on a surface.
Example 9
Let be a compact, orientable surface and let
be an embedded simple closed curve that is not homotopic to a point. Let

be a finite-sheeted covering map. The pre-image is easily seen to be a multicurve

and by the homotopy lifting property no is homotopic to a point.
Now let us translate this into group theory. The curve decomposes
as either an amalgamated free product or an HNN extension (depending on whether
is separating or non-separating). But the multicurve
decomposes
as a graph of groups. Of course we could focus on just one of the
, which decomposes
as an amalgamated product or HNN extension, but that would involve making an unnatural choice. Furthermore, the multicurve
has the attractive property that the restriction of the covering map
to a connected component of
is a covering map onto a connected component of
.
Example 9 suggests that a subgroup of an amalgamated product or HNN extension naturally decomposes as a graph of groups. To be precise, the following is true for amalgamated products, and a similar theorem holds for HNN extensions (or indeed any graph of groups).







The most usual modern technique for proving theorems about graphs of groups (including amalgamated products and HNN extensions) makes use of Serre's observation that every graph of groups corresponds to an action of its fundamental group on a tree. This tree is called the Bass–Serre tree, and is the object of study in Bass–Serre Theory.
Comments
Tensor products are not
Sat, 25/04/2009 - 04:55 — emertonTensor products are not really defined for groups, but rather for modules
-modules, and so tensor products are defined for abelian groups, but this is a construction of a very different flavour
over rings. Abelian groups are
to all the other constructions listed on this page.
Perhaps it would be better to have a comment somewhere on the page to this effect
(i.e. that one can define the tensor product of two abelian groups), and then just
link to the How to use tensor products page for more details.
If there are no objections, I will do this some time soon.
I agree
Sat, 25/04/2009 - 08:28 — JoseBroxYes, I was thinking about the tensor product for abelian groups as a special case of a product construction (in Ring theory it is quite usual to think of everything as modules). Feel free to change it as you say, I added it just as a suggestion (I put it on the list because there really isn't any more on the stub at the moment!)
There could be a link
Mon, 11/05/2009 - 04:11 — emertonThere could be a link somewhere among the later examples to Use topology to study your group, although I haven't thought very carefully about where it would sit best.