We show that the Boolean Prime Ideal Theorem () does not imply the Nielsen‐Schreier Theorem () in, thus strengthening the result of Kleppmann from “Nielsen‐Schreier and the Axiom of Choice” that the (strictly weaker than ) Ordering Principle () does not imply in. We also show that is false in Mostowski's Linearly Ordered Model of. The above two results also settle the corresponding open problems from Howard and Rubin's “Consequences of the Axiom of Choice”.

The Nielsen‐Schreier theorem asserts that subgroups of free groups are free. In the first section we show that this theorem does not follow from the Linear Ordering Principle, thus strengthening the fact that it implies the Axiom of Choice for families of finite sets. In the second section, we show that a stronger variant of the Nielsen‐Schreier theorem implies the Axiom of Choice.