One of the most fundamental notions from abstract algebra is that of a group. A group is a (possibly infinite) set of elements together with multiplication and inverse operators that satisfy certain simple axioms. It turns out that there are some interesting connections between infinite groups and theoretical computer science -- computability and automata theory, in particular.
In this talk I will discuss some solved and unsolved problems that arise in this context. No previous knowledge of group theory will be assumed, and while some previous exposure to decidability, formal languages and automata will be helpful, it will not be necessary.