Explanation of Term
Nilpotent groups are so called because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group G of nilpotence degree n and an element g, the function defined by (where is the commutator of g and x) is nilpotent in the sense that the nth iteration of the function is trivial: for all in .
This is not a defining characteristic of nilpotent groups: groups for which is nilpotent of degree n (in the sense above) are called n-Engel groups, and need not be nilpotent in general. They are proven to be nilpotent if they have finite order, and are conjectured to be nilpotent as long as they are finitely generated.
An abelian group is precisely one for which the adjoint action is not just nilpotent but trivial (a 1-Engel group).
Read more about this topic: Nilpotent Group
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