# Nilpotent Lie rings of prime-cube order

## 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 understand how to 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