We suggest a cohomological framework to describe groups of I-type and involutive Yang-Baxter groups. These groups are key in the study of involutive non-degenerate set-theoretic solutions of the quantum Yang-Baxter equation. Our main tool is a lifting criterion for 1-cocycles, established here in a general non-abelian setting.

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Abstract. The analogue of Lagrangians for symplectic forms over finite groups is studied, motivated by the fact that symplectic G-forms with a normal Lagrangian N ⊳G are in one-to-one correspondence, up to inflation, with bijective 1-cocycle data on the quotients G/N . This yields a method to construct groups of central type from such quotients, known as Involutive Yang-Baxter groups. Another motivation for the search of normal Lagrangians comes from a noncommutative generalization of Heisenberg liftings which require normality.Although it is true that symplectic forms over finite nilpotent groups always admit Lagrangians, we exhibit an example where none of these subgroups is normal. However, we prove that symplectic forms over nilpotent groups always admit normal Lagrangians if all their p-Sylow subgroups are of order less than p 8 .