Lazard Lie ring

Definition

A Lazard Lie ring is a Lie ring $L$ such that there exists a natural number $c$ with both the following two properties:

No.	Shorthand for property	Explanation
1	The additive group is a powered group for the set of all primes less than or equal to $c$.	For any prime number $p\leq c$, and any element $a\in L$, there is a unique element $b\in L$ such that $pb=a$.
2	The 3-local nilpotency class is at most $c$.	For any subset of $L$ of size at most three, the subring of $L$ generated by that subset is a nilpotent Lie ring of nilpotency class at most $c$.

Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as $c$ increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase $c$. Thus, a particular value of c may work for a Lie ring but larger and smaller values may not.

A Lazard Lie ring is a Lie ring that can participate as the Lie ring side of a Lazard correspondence. The other side of the correspondence is its Lazard Lie group.

p-Lie ring

A Lazard $p$-Lie ring is a special case of the above, namely a Lie ring $L$ such that:

1. There is a prime $p$ such that every element of $L$ has order a power of $p$.
2. The Lie subring of $L$ generated by any three elements of $L$ is a nilpotent Lie ring of nilpotency class less than $p$.

A Lazard $p$ Lie ring is a $p$-Lie ring that can participate as the Lie ring side of a Lazard correspondence. The other side of the correspondence is its Lazard Lie group, which is a p-group.

Relation with other properties

Stronger properties

• Abelian Lie ring

Weaker properties

• Nilpotent Lie ring