## How to determine whether a finitely presented group is finite

### Quick description

There is no general algorithm that will determine whether a given finitely presented group is finite or not. However, there are a number of strategies to try.

### Prerequisites

Basic definitions of combinatorial group theory.

### General discussion Contributions wanted This article could use additional contributions. I have another idea or two to add, but other strategies would be greatly appreciated!

Suppose that is a finitely presented group. It is easy to find quotients of ; adding relations to the presentation and determining the consequences amounts to passing to a quotient of by the normal closure of the relations. Then if you can add relations to and end up with a group that you know to be infinite, it follows that the original group was infinite.

### Example 1

Let be the finitely presented group with presentation . Trying to determine explicitly whether there are infinitely many reduced words in seems hopelessly complicated. Instead, let's notice that if we add the relation , then this group collapses to , which we recognize as the infinite Coxeter group .