# Classification of nilpotent Lie rings of prime-cube order

## Contents

## Statement

Let be a prime number. There are, up to isomorphism, *five* possibilities for a nilpotent Lie ring of order . Three of them are abelian and two are non-abelian.

For odd primes, we can use the Baer correspondence (a special case of the Lazard correspondence) to achieve a one-to-one correspondence between the nilpotent Lie rings of order and the groups of order .

For the prime , although the *number* of nilpotent Lie rings equals the number of groups, we cannot use the Baer correspondence. More specifically, there is no natural way of matching the two non-abelian groups of order and the two non-abelian nilpotent Lie rings of order .

### The three abelian Lie rings

The three abelian Lie rings are the Lie rings with trivial Lie bracket; these thus correspond to the abelian groups of order . They are classified by the partitions of 3:

Partition of 3 | Corresponding additive group of Lie ring |
---|---|

3 | cyclic group of prime-cube order, denoted or , or |

2 + 1 | direct product of cyclic group of prime-square order and cyclic group of prime order, denoted or |

1 + 1 + 1 | elementary abelian group of prime-cube order, denoted , or , or |

### The two non-abelian nilpotent Lie rings

The two non-abelian nilpotent Lie rings are given as follows:

Lie ring name | Presentation | Nilpotency class | Additive group | Corresponding group under Baer correspondence in case of odd prime |
---|---|---|---|---|

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 | semidirect product of cyclic group of prime-square order and cyclic group of prime order | |

upper-triangular nilpotent matrix Lie ring:u(3,p) | 2 | elementary abelian group of prime-cube order | prime-cube order group:U(3,p); see Baer correspondence between u(3,p) and U(3,p) |

## Proof

### First part of proof: crude description of center and quotient by center

**Given**: A prime number , a nilpotent Lie ring of order .

**To prove**: Either is abelian, or is cyclic of order and the quotient is an abelian Lie ring of order whose additive group is elementary abelian of order .

**Proof**: Let be the center of .

*Note: Some proof details need to be clarified, but the outline is complete and good enough for people thorough with group theory.*

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | is nontrivial | is nilpotent and nontrivial | Given direct | ||

2 | The order of cannot be | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
has order | ||

3 | The order of is either or | Steps (1), (2) | |||

4 | If has order , then is cyclic of order and the quotient is elementary abelian of order . | ||||

5 | If has order , is abelian | ||||

6 | We get the desired result. | Steps (4), (5) | Step-combination. |

### Second part of proof: classifying the abelian Lie rings

### Third part of proof: classifying the non-abelian nilpotent Lie rings

Since we already know and , we need to specify two things:

- The alternating bilinear map given by the Lie bracket: There is only one option for this (up to isomorphism) because is isomorphic to , and, up to conjugacy, there is a unique isomorphism.
- The nature of the additive group extension of on top of : There are two possibilities for this: elementary abelian group of prime-square order (corresponding to the trivial or split extension)and direct product of cyclic group of prime-square order and cyclic group of prime order (corresponding to the nontrivial extensions).

Working out the corresponding Lie ring structures, we get upper-triangular nilpotent matrix Lie ring:u(3,p) (corresponding to the trivial extension) and semidirect product of cyclic Lie ring of prime-square order and cyclic Lie ring of prime order (corresponding to the nontrivial extension).