We discuss the cardinalities of maximal cofinitary groups under the assumption of $\neg CH$ . We also discuss various open questions in this area.

We show that it is consistent with ZFC + ¬CH that there is a maximal cofinitary group (or, maximal almost disjoint group) G ≤ Sym(ω) such that G is a proper subset of an almost disjoint family A $\subseteq$ Sym(ω) and |G| < |A|. We also ask several questions in this area.

Assuming Martin's Axiom, we compute the value of the cofinality of the symmetric group on the natural numbers. We also show that Martin's Axiom does not decide the value of the covering number of a related Mycielski ideal.

We show that g ≤ c(Sym(ω)) where g is the groupwise density number and c(Sym(ω)) is the cofinality of the infinite symmetric group. This solves (the second half of) a problem addressed by Thomas.