# Nilpotent Lie rings of prime-cube order

From Groupprops

## Statistics at a glance

Quantity | Value |
---|---|

Total number of nilpotent Lie rings | 5 |

Number of abelian Lie rings | 3 |

Number of non-abelian Lie rings of class two | 2 |

To understandhowto come up with this list of nilpotent Lie rings and show that it is exhaustive, see classification of nilpotent Lie rings of prime-cube order

## The list

Common name for Lie ring | Nilpotency class | Isomorphism class of additive group |
---|---|---|

cyclic Lie ring of prime-cube order | 1 | cyclic group of prime-cube order |

abelian Lie ring whose additive group is direct product of cyclic group of prime-square order and cyclic group of prime order | 1 | direct product of cyclic group of prime-square order and cyclic group of prime order |

abelian Lie ring whose additive group is elementary abelian group of prime-cube order | 1 | elementary abelian group of prime-cube order |

upper-triangular nilpotent matrix Lie ring:u(3,p) | 2 | elementary abelian group of prime-cube order |

semidirect product of cyclic Lie ring of prime-square order and cyclic Lie ring of prime order | 2 | direct product of cyclic group of prime-square order and cyclic group of prime order |