ar X iv : m at h / 05 08 14 6 v 1 [ m at h . G N ] 8 A ug 2 00 5 THE COMBINATORICS OF THE BAIRE GROUP

We study subgroups of Z which possess group theoretic properties analogous to properties introduced by Menger (1924), Hurewicz (1925), Rothberger (1938), and Scheepers (1996). We obtain purely combinatorial characterizations of these properties, and combine them with other techniques to solve several questions of Babinkostova, Kočinac, and Scheepers. We also show that, while the Menger Conjecture, when formulated for metrizable groups, is false, the corresponding version of the Hurewicz Conjecture is true. This achieves a goal set by Okunev to find an inner characterization of subgroups of σ-compact groups.