You might know the theorem of Lagrange from group theory: A finite group of order n can have a subgroup of order m only if m divides n. Here the order is just the number of elements, a natural number.

Recently I came across a generalization to profinite groups. How do you make sense of the order of an infinite group? How to say that the order of a subgroup divides the order of the group? The solution is a simple concept called supernatural numbers, which I will explain in this short article. The main part should in principle be accessible to non-mathematicians as well.

Continue reading «Lagrange and the Supernaturals»

I recently learned how to build a Haar measure on every locally compact group. It's a fact there is only one (up to positive scalar multiple) Haar measure on a locally compact group, and it's easy to see that Lie groups (which includes algebraic and finite groups) and all compact groups are locally compact, so they have a unique (up to scalar multiple) Haar measure, too.
But the Haar measure can be defined much easier for Lie groups, and it's even simpler for finite groups. I wanted to study the relation more directly than by the uniqueness proof one sees in the literature.
This text is intended to be read by anyone who is familiar with the notion of groups and measures. Maybe you will want to consult Wikipedia along the lines - I have included some links.

I give first a precise definition of Haar measure and a state its uniqueness on locally compact groups, then I compare the different types of topological groups I want to investigate, along with valid definitions of Haar measure.
Continue reading «Haar measure in different settings»