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Isotropy in group cohomology. (English) Zbl 1330.20015

Summary: The analog of Lagrangians for symplectic forms over finite groups is studied, motivated by the fact that symplectic \(G\)-forms with a normal Lagrangian \(N\triangleleft G\) 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 non-commutative generalization of Heisenberg liftings that 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 \(p^8\).

MSC:

20C25 Projective representations and multipliers
20J06 Cohomology of groups
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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