Exploring Amalgamated Products in Group Theory: A Brief Overview

[MathematicalPhysicist]

i found in this paper the term in the title:
http://arxiv.org/abs/math.GR/9305201
what can you tell me about them?
i didnt find anything about them in mathworld.com

the only thing i know from the paper is that it concerns group theory.
(is this the right forum for this kind of question? if not move it as you please).

 

[matt grime]

Let H and K be two groups with maps from a third group G to each, then the amalgamated product over G (with these maps) often denoted H x_G K is the pushout of the diagram in Grp, or if you don't like category theory it is an object (group) with maps from H and K such that the composites with the injections from G agree, and it is universal with this property.

this exists and is unique up to unique isomorphism, as long as your groups have a set of generators.

examples of this abound in topology where it is used to find the homotopy groups using the Seifert-Van Kampen theorem. (some people don't use the Seifert part of the name)

example compute the fundamental group of the torus:

we cover with two open patches one is just the torus less a point (this is homotopic to the bouquet of two circles), the other a small open disc about this point. the over lap is then homotopic to a circle, thus we get

G= Z, H=triv, K=Free prod on two gens=F_2, the fundamental groups of those subset.

the single cycle in G is sent to the loop xyx^{-1}y^{-1}, and the id in triv, so these must be identified in the amalgamated porduct, and that is the only rule we see the generators must satisify, so that just tells us to abelianize F_2, ie ZxZ.

 

[tacman]

Yes, this is the right forum for this type of question. Amalgamated products in group theory are a concept that arises from the study of group theory, which is a branch of abstract algebra. In a nutshell, an amalgamated product is a construction that allows us to combine two or more groups in a specific way to create a new group. This construction is useful in studying the structure of groups and their subgroups.

In more technical terms, an amalgamated product is a generalization of the free product of two groups, where we identify certain elements between the two groups. This construction is often used in the study of group presentations and the classification of groups.

In the paper you mentioned, the authors provide a brief overview of amalgamated products and their properties. They also discuss some applications of amalgamated products in group theory, such as in the study of amalgamated free products and HNN extensions.

It is not surprising that you did not find much information about amalgamated products on mathworld.com, as it is a relatively specialized topic in group theory. However, there are many other resources available online and in textbooks that discuss amalgamated products in more detail. So if you are interested in learning more about this topic, you can certainly find more information with a little bit of research.