# Lie rings of prime-cube order

## The list

Let $p$ be a prime number. Up to isomorphism, here are the Lie rings of order $p^3$ (the table is not yet complete):

No. Lie ring definition Nilpotent? Additive group Partition corresponding to additive group
1 abelian Lie ring whose additive group is the cyclic group of prime-cube order Yes cyclic group of prime-cube order $3$
2 abelian Lie ring whose additive group is the direct product of cyclic group of prime-square order and cyclic group of prime order Yes direct product of cyclic group of prime-square order and cyclic group of prime order $2 + 1$
3 semidirect product of cyclic Lie ring of prime-square order and cyclic Lie ring of prime order, i.e., Lie ring given by presentation $\langle a,b \mid p^2a = pb = 0, [a,b] = pa \rangle$ Yes direct product of cyclic group of prime-square order and cyclic group of prime order $2 + 1$
4 nontrivial semidirect product of cyclic Lie ring of prime order and cyclic Lie ring of prime-square order, i.e., Lie ring given by presentation $\langle a,b \mid p^2a = pb = 0, [a,b] = b \rangle$ No direct product of cyclic group of prime-square order and cyclic group of prime order $2 + 1$
5 abelian Lie ring whose additive group is the elementary abelian group of prime-cube order Yes elementary abelian group of prime-cube order $1 + 1 + 1$
6 upper-triangular nilpotent matrix Lie ring:u(3,p), i.e., Lie ring given by the presentation $\langle a,b,c \mid pa = pb = pc = 0, [a,c] = b, [a,b] = [b,c] = 0 \rangle$ Yes elementary abelian group of prime-cube order $1 + 1 + 1$
7 direct product of cyclic Lie ring of prime order and nontrivial semidirect product of Lie rings of prime order, i.e., Lie ring given by the presentation $\langle a,b,c \mid pa = pb = pc = 0, [a,b] = a, [a,c] = [b,c] = 0 \rangle$ No elementary abelian group of prime-cube order $1 + 1 + 1$

There are some more to be filled in

For the explanation of why these are precisely the Lie rings of prime-cube order, see classification of Lie rings of prime-cube order. If you are interested in only the nilpotent ones, see classification of nilpotent Lie rings of prime-cube order.

## Baer correspondence for odd primes

If $p$ is an odd prime, then there is a correspondence between the nilpotent Lie rings among the above and the groups of prime-cube order, given below: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]