Inner-Lazard Lie group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

Definition

Quick definition

A group is termed a Lazard Lie group if its 3-local nilpotency class is finite and less than or equal to (the group's powering threshold + 1).

Explicit definition

A group G is termed an inner-Lazard Lie group if there is a natural number c such that both the following hold:

No. Shorthand for property Explanation
1 G is powered for the set of all primes strictly less than c. G is uniquely p-divisible for all primes p < c. In other words, if p \le c is a prime and g \in G, there is a unique value h \in G satisfying h^p = g.
2 The 3-local nilpotency class of G is at most c. For any three elements of G, the subgroup of G generated by these three elements is a nilpotent group 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 group but larger and smaller values may not.

p-group version

A p-group is termed an inner-Lazard Lie group if its 3-local nilpotency class is at most p. In other words, every subgroup of it generated by at most three elements has nilpotency class at most p where p is the prime associated with the group.