Isomorphism types of definable (maximal) cofinitary groups

Abstract: 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.