The aim of this paper is to study the concatenation hierarchy whose level 0 consists of all group languages. The union of all the levels of this hierarchy is the closure of group languages under product and boolean operations. Our first result states that this union is a decidable variety of languages. The rest of the paper is devoted to the study of level 1. This variety of languages, and the corresponding variety of monoids ◊G, appear in many different contexts. First, ◊G is exactly the variety J * G generated by all semidirect products of a J-trivial monoid by a group. Second ◊G is also the variety generated by powermonoids of groups. Finally, languages of level 1 arise in the study of the finite group topology for the free monoid. An important problem is to know whether this variety is decidable. The discussion of this problem motivated the introduction of the variety of monoids BG, which is the variety of all monoids M such that, for every idempotent e, f of M, efe = e implies e = f. Several equivalent definitions are given in the paper. In particular, we show that a monoid M is in BG if and only if the submonoid generated by all idempotents of M belongs to J. We also prove that BG is generated by all monoids that are, in some sense, extensions of a group by a monoid of J. Finally, ◊G is contained in BG but we don't know if this inclusion is strict or not.