Nilpotent group: Difference between revisions

{{Short description|GroupConcept thatin hasgroup antheory upperof central series terminating with Gmathematics}}

Nilpotent groups are so called so because the "adjoint action" of any element is [[nilpotent]], meaning that for a nilpotent group <math>G</math> of nilpotence degree <math>n</math> and an element <math>g</math>, the function <math>\operatorname{ad}_g \colon G \to G</math> defined by <math>\operatorname{ad}_g(x) := [g,x]</math> (where <math>[g,x]=g^{-1} x^{-1} g x</math> is the [[commutator]] of <math>g</math> and <math>x</math>) is nilpotent in the sense that the <math>n</math>th iteration of the function is trivial: <math>\left(\operatorname{ad}_g\right)^n(x)=e</math> for all <math>x</math> in <math>G</math>.