Inner-Lazard Lie group
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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A group is termed an inner-Lazard Lie group if there is a natural number such that both the following hold:
|No.||Shorthand for property||Explanation|
|1||is powered for the set of all primes strictly less than .||is uniquely -divisible for all primes . In other words, if is a prime and , there is a unique value satisfying .|
|2||The 3-local nilpotency class of is at most .||For any three elements of , the subgroup of generated by these three elements is a nilpotent group of nilpotency class at most .|
Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase . Thus, a particular value of may work for a group but larger and smaller values may not.
A p-group is termed an inner-Lazard Lie group if its 3-local nilpotency class is at most . In other words, every subgroup of it generated by at most three elements has nilpotency class at most where is the prime associated with the group.