### Abstract

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 p8.

Original language | English |
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Pages (from-to) | 587-599 |

Number of pages | 13 |

Journal | Bulletin of the London Mathematical Society |

Volume | 46 |

Issue number | 3 |

Early online date | 1 Apr 2014 |

DOIs | |

Publication status | Published - Jun 2014 |

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### Cite this

*Bulletin of the London Mathematical Society*,

*46*(3), 587-599. https://doi.org/10.1112/blms/bdu018

**Isotropy in group cohomology.** / Ben David, Nir; Ginosar, Yuval; Meir, Ehud.

Research output: Contribution to journal › Article

*Bulletin of the London Mathematical Society*, vol. 46, no. 3, pp. 587-599. https://doi.org/10.1112/blms/bdu018

}

TY - JOUR

T1 - Isotropy in group cohomology

AU - Ben David, Nir

AU - Ginosar, Yuval

AU - Meir, Ehud

N1 - We thank P. Singla and U. Onn for their helpful suggestions.

PY - 2014/6

Y1 - 2014/6

N2 - 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?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 p8.

AB - 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?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 p8.

U2 - 10.1112/blms/bdu018

DO - 10.1112/blms/bdu018

M3 - Article

VL - 46

SP - 587

EP - 599

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

IS - 3

ER -