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Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in 
$\mathsf {ZF}$, i.e., without using the Axiom of Choice) maximal cofinitary groups. Among the groups we construct, one is definable by a formula in second-order arithmetic with only a few natural number quantifiers.
Assuming that every set is constructible, we find a 
${\text{\Pi }}_1^1 $ maximal cofinitary group of permutations of 
$\mathbb{N}$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a 
${\text{\Pi }}_1^1 $ maximal cofinitary group in L.
