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Concatenation hierarchies are classifications of regular languages. All suchhierarchies are built through the same construction process: start from aninitial class of languages and build new levels using two generic operations.Concatenation hierarchies have gathered a lot of interest since the 70s, thanksto an alternate logical definition: each concatenation hierarchy is thequantification alternation hierarchy within a variant of first-order logic overwords.  Our goal is to understand such hierarchies. We look at two decision problems:membership and separation. For a class of languages C, C-separation takes tworegular languages as input and asks whether there exists a third one in Cincluding the first one and disjoint from the second one. Settling whetherseparation is decidable for the levels within a given concatenation hierarchyis among the most fundamental and challenging questions in formal languagetheory. In all prominent cases, it is open, or answered positively for lowlevels only. Recently, a breakthrough was made using a generic approach for aspecific kind of hierarchy: those with a finite basis. In this case, separationis always decidable for levels 1/2, 1 and 3/2.  Our main theorem is similar but independent: we consider hierarchies withpossibly infinite bases, but that may only contain group languages. An exampleis the quantifier alternation hierarchy of first-order logic with modularpredicates: its basis consists of languages counting the length of words modulosome number. Using a generic approach, we show that for any such hierarchy, ifseparation is decidable for the basis, then it is decidable for levels up to3/2. This complements the aforementioned result nicely: all bases considered inthe literature are either finite or made of group languages. Thus, one mayhandle the lower levels of any prominent hierarchy in a generic way.

Separation and covering for group based concatenation hierarchies