Local nilpotency class

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Suppose G is a group and k is a natural number. The k-local nilpotency class is defined as the supremum, over all subgroups H of G generated by sets of size at most k, of the nilpotency class of H. In other words, it is defined as:

\sup_{S \subseteq G, |S| \le k} \operatorname{class}(\langle S \rangle)

If there is a non-nilpotent subgroup of G generated by k or fewer elements, then the k-local nilpotency class is \infty. The k-local nilpotency class may also be infinite because, even though all the subgroups generated by at most k elements are nilpotent, there is no finite upper bound on their nilpotency class.

Particular cases