A new characterization for $p$-local balanced projective groups

By introducing the notion of a ${\text{K}}$-nice submodule, we obtain a characterization of $p$-local balanced projectives perfectly analogous to the familiar third axiom of countability characterization of totally projective $p$-groups. We use this new characterization to prove that if a $p$-local group $G$ satisfies the third axiom of countability with respect to nice submodules and has a ${\text{K}}$-basis, then $G$ is a balanced projective.