Abstract:A torsion-free group is a $B_2$-group if and only if it has an axiom-3 family $\frak C$ of decent subgroups such that each member of $\frak C$ has such a family, too. Such a family is called $SL_{\aleph _0}$-family. Further, a version of Shelah's Singular Compactness having a rather simple proof is presented. As a consequence, a short proof of a result [R1] stating that a torsion-free group $B$ in a prebalanced and TEP exact sequence $0 \to K \to C \to B \to 0$ is a $B_2$-group provided $K$ and $C$ are so.
Keywords: $B_1$-group, $B_2$-group, prebalanced subgroup, torsion extension property, decent subgroup, axiom-3 family
AMS Subject Classification: 20K20