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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.


Basic definitions of combinatorial group theory.

General discussion

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

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

Example 1

Let G be the finitely presented group with presentation \langle a, b, c, d \mid ada, a^2, (bd)^2, c^2, d^4, (abdadb)^2, (bc)^4, (acd)^2 \rangle. Trying to determine explicitly whether there are infinitely many reduced words in G seems hopelessly complicated. Instead, let's notice that if we add the relation d = 1, then this group collapses to \langle a, b, c \mid a^2, b^2, c^2, (ab)^4, (bc)^4, (ac)^2 \rangle, which we recognize as the infinite Coxeter group [4,4].


methods to show that a group is infinite

I have a couple of posts on my blog ( with lots of examples of ways to show that a group is finite/infinite, probably too many/too specialized for the tricki. You are welcome to mine these posts for anything you think is relevant.