Algebra Seminar talk
Maximal Cofinitary Groups
A subgroup of $S_\infty$, the group of all permutations of the natural numbers, is said to be cofinitary, if all of its non-identity elements have only finitely many fixed points. In 1988 Adeleke showed that every countable cofinitary group is a proper subgroup of a highly transitive cofinitary group. Thus in particular maximal cofinitary groups are uncountable, which initiated the study of the spectrum of possible (infinite) sizes of maximal cofinitary groups.
In addition, we will consider the relationship between cofinitary groups and other combinatorial objects on the real line. We will give an outline of a result of Brendle, Spinas and Zhang, which states that the minimal size of a maximal cofinitary group is not smaller than the minimal size of a non-meager set. The result implies an important distinction between maximal cofinitary groups and some of its close combinatorial relatives.
We will conclude with a brief discussion of open questions.