This document summarizes the key points from a talk on locally finite maximal cofinite groups. It begins with definitions of cofinite permutations and cofinite and maximal cofinite groups. It then discusses previous results on the complexity of maximal almost disjoint families and maximal cofinite groups. The main result is that under the assumption of Martin's Axiom, there exists a locally finite maximal cofinite group. The proof constructs such a group by carefully adding finite permutations to an increasing union of finite cofinite groups.