Classification of nilpotent Lie rings of prime-cube order

Statement

Let $p$ be a prime number. There are, up to isomorphism, five possibilities for a nilpotent Lie ring of order $p^{3}$. 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 $p^{3}$ and the groups of order $p^{3}$.

For the prime $p=2$, 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 $p^{3}$ and the two non-abelian nilpotent Lie rings of order $p^{3}$.

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 $p^{3}$. 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 $C_{p^{3}}$ or $\mathbb {Z} _{p^{3}}$, or $\mathbb {Z} /p^{3}\mathbb {Z}$
2 + 1	direct product of cyclic group of prime-square order and cyclic group of prime order, denoted $C_{p^{2}}\times C_{p}$ or $\mathbb {Z} _{p^{2}}\times \mathbb {Z} _{p}$
1 + 1 + 1	elementary abelian group of prime-cube order, denoted $E_{p^{3}}$, or $C_{p}\times C_{p}\times C_{p}$, or $\mathbb {Z} _{p}\times \mathbb {Z} _{p}\times \mathbb {Z} _{p}$

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 $p$
semidirect product of cyclic Lie ring of prime-square order and cyclic Lie ring of prime order	$\langle a,b\mid p^{2}a=pb=0,[a,b]=a\rangle$	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)	$\langle a,b,c\mid pa=pb=pc=0,[a,c]=b,[a,b]=[b,c]=0\rangle$	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 $p$, a nilpotent Lie ring $P$ of order $p^{3}$.

To prove: Either $P$ is abelian, or $Z(P)$ is cyclic of order $p$ and the quotient is an abelian Lie ring of order $p^{2}$ whose additive group is elementary abelian of order $p^{2}$.

Proof: Let $Z=Z(P)$ be the center of $P$.

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	$Z$ is nontrivial		$P$ is nilpotent and nontrivial		Given direct
2	The order of $Z$ cannot be $p^{2}$	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	$P$ has order $p^{3}$
3	The order of $Z$ is either $p$ or $p^{3}$			Steps (1), (2)
4	If $Z$ has order $p$, then $Z$ is cyclic of order $p$ and the quotient $P/Z$ is elementary abelian of order $p^{2}$.
5	If $Z$ has order $p^{3}$, $P$ is abelian
6	We get the desired result.			Steps (4), (5)	Step-combination.

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

Since we already know $Z(P)$ and $P/Z(P)$, we need to specify two things:

• The alternating bilinear map $P/Z(P)\times P/Z(P)\to Z(P)$ given by the Lie bracket: There is only one option for this (up to isomorphism) because $\bigwedge ^{2}(P/Z(P))$ is isomorphic to $Z(P)$, and, up to conjugacy, there is a unique isomorphism.
• The nature of the additive group extension of $P/Z(P)$ on top of $Z(P)$: 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).