**Def.** A group language is a recognizable language whose syntactic monoid is a group.


**q1.** Is it known a "nice" combinatorial characterization of group languages ?


**q1.1.** If no, is it well understood why it is so? Is there recent progress on this ?


**q2.** Let F be a proper family of group languages (for example, abelian groups, etc...)
Do there exist notable instances of F such that a nice combinatorial characterization of F-languages arise ? 



Progress on group languages characterizations
Usually it is deemed impossible to describe group languages completely since this involves understanding the word problem for each finite simple group for any generating set. 


Nice descriptions are known for subclasses like p-groups, nilpotent groups, supersolvable groups and solvable groups. p-groups are done in Eilenberg's book. Nilpotent groups follow from p-groups but Therien refined this to nilpotent groups of class c. Straubing did solvable groups. Supersolvable groups was published by Pin and collaborators. 


