Lazard Lie ring

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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 \le 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

Weaker properties