Classification of nilpotent Lie rings of prime-cube order
- 1 Statement
- 2 Proof
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)|
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).