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
Famous quotes containing the words explanation of, explanation and/or term:
“We live between two worlds; we soar in the atmosphere; we creep upon the soil; we have the aspirations of creators and the propensities of quadrupeds. There can be but one explanation of this fact. We are passing from the animal into a higher form, and the drama of this planet is in its second act.”
—W. Winwood Reade (1838–1875)
“Auden, MacNeice, Day Lewis, I have read them all,
Hoping against hope to hear the authentic call . . .
And know the explanation I must pass is this
MYou cannot light a match on a crumbling wall.”
—Hugh MacDiarmid (1892–1978)
“Most literature on the culture of adolescence focuses on peer pressure as a negative force. Warnings about the “wrong crowd” read like tornado alerts in parent manuals. . . . It is a relative term that means different things in different places. In Fort Wayne, for example, the wrong crowd meant hanging out with liberal Democrats. In Connecticut, it meant kids who weren’t planning to get a Ph.D. from Yale.”
—Mary Kay Blakely (20th century)