Kastermans proved that consistently $\bigoplus_{ \aleph_1} \mathbb{ Z}_2$ has a cofinitary representation. We present a short proof that $\bigoplus_{ \mathfrak{ c}} \mathbb{ Z}_2$ always has an arithmetic cofinitary representation. Further, for every finite group $F$ we construct an arithmetic maximal cofinitary group of isomorphism type $(\ast_{ \mathfrak{ c}} \mathbb{ Z}) \times F$. This answers an implicit question by Schrittesser and Mejak whether one may construct definable maximal cofinitary groups not decomposing into free products.
Categories: math.LO