Isotropy in Group Cohomology

TL;DR

This paper explores the structure of symplectic forms over finite groups, establishing conditions for the existence of normal Lagrangians and introducing methods to construct groups of central type using cocycle data.

Contribution

It provides new insights into the existence of normal Lagrangians in symplectic forms over nilpotent groups and links symplectic forms to group construction via cocycle data.

Findings

Symplectic forms over finite nilpotent groups may lack normal Lagrangians.

Normal Lagrangians exist if all p-Sylow subgroups are smaller than p^8.

A method to construct groups of central type from quotient data is proposed.

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 non-commutative 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.