Constructing maximal cofinitary groups

TL;DR

This paper presents a choice-free method to construct maximal cofinitary groups, including a definable example with minimal quantifiers, advancing understanding in group theory and logic.

Contribution

It improves and clarifies a previous construction, demonstrating how to build maximal cofinitary groups within ZF without the Axiom of Choice, and provides a definable example.

Findings

Constructed maximal cofinitary groups in ZF without Choice.

Provided a definable maximal cofinitary group with few quantifiers.

Clarified the construction method of Horowitz and Shelah.

Abstract

Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in $\textsf{ZF}$, that is, 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.