# Amalgams or A Tale About Two Groups Who Wish To Become One

## Document Type

Presentation Abstract

## Presentation Date

12-12-2002

## Abstract

Webster's Ninth Collegiate Dictionary defines an *amalgam* as a mixture of mercury and another metal, or more generally as a mixture of two different elements. Dental fillings, for example, are made of amalgams.

Mathematically, an *amalgam* is obtained when we take two structures (two topological spaces, two graphs, two rings), and mix them by identifying some subparts of them; The challenge then is to find a bigger structure where this mixture can live.

For example, if we have two groups *G* and *K*, and we have a subgroup *H* which is common to both, we want to think of *G* and *K* as being "glued together along *H*;" this is an amalgam of the two groups. It is not a group, because there is no way to multiply an element of *G* which is not in *H* by an element of *K* which is not in *H*. What we want is to find some larger group *M*, which contains *G* and *K* as subgroups in such a way that their intersection is still *H*. Similar problems exist if we replace "group" and "subgroup" with "topological space" and "topological subspace"; or with "graph" and "subgraph"; or with "manifold" and "submanifold", etc.

This simple question leads very easily to some very powerful mathematics, and of course to even more questions, many of which we cannot yet answer. I will give a tour of amalgams of groups, assuming nothing more than a basic knowledge of them. As opposed to getting to know what a dental amalgam is, it will not hurt at all.

## Recommended Citation

Magidin, Dr. Arturo, "Amalgams or A Tale About Two Groups Who Wish To Become One" (2002). *Colloquia of the Department of Mathematical Sciences*. 131.

https://scholarworks.umt.edu/mathcolloquia/131

## Additional Details

Thursday, 12 December 2002

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)