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$B_1$-group; $B_2$-group; prebalanced subgroup; torsion extension property; decent subgroup; axiom-3 family
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.
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[R2] Rangaswamy K.M.: A property of $B_2$-groups. Proc. Amer. Math. Soc. 121 (1994), 409-415. MR 1186993
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