# Historical reference request on Nilpotent groups

From Wikipedia:

"Abelian groups were named after Norwegian mathematician Niels Henrik Abel by Camille Jordan because Abel found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using radicals"

"Historically, the word solvable arose from Galois theory and the proof of the general unsolvability of quintic equation."

Question. Where do nilpotent groups come from ? Who defined them first ? For what purpose ? Any reference ?
• btw the original French word for solvable was résoluble and has two English translations in use: the most common solvable and also soluble
– YCor
Jun 2 '16 at 17:38
• Isn't there one which is more British English and the other American English ? Jun 2 '16 at 17:41
• Yes soluble is more usual in British and solvable in America although, despite being British, I prefer solvable myself. I associate soluble more with chemistry (salt is soluble in water). Jun 2 '16 at 17:51
• It's not named after Niels Nilpotent. The term “nilpotent group” is based on an analogy with ring theory. In a ring, an element with a power equal to 0 is called nilpotent. In a Lie algebra $\mathfrak g$, an element $x$ could be called nilpotent if the linear operator $y \mapsto [x, y]$ on $\mathfrak g$ is nilpotent in the sense of ring theory (a power of the operator is 0). By a theorem of Engel, all elements of a Lie algebra are nilpotent if and only if the Lie group corresponding to the Lie algebra has the property we call being a nilpotent group. Jun 2 '16 at 18:07
• This post seems to be better suited to History of Science and Mathematics, since it mainly historical in nature.
– Danu
Jun 3 '16 at 12:17

In 1870, the American mathematician, Benjamin Pierce first introduced the term nilpotent in the context of his work on the classification of Algebras. In Algebra, an element $x$ of a ring $R$ is said to be nilpotent if there exists some positive integer $n$ such that $x^{n}=0$.
In group theory, a nilpotent group is a group having a special property that makes it 'almost abelian' through repeated application of the commutator operation defined by $[x,y]=xyx^{-1}y^{-1}$
For justification of the term nilpotent, start with a nilpotent group $G$ and an element $g$ of $G$ and define a function $f : G \longrightarrow G$ by $f(x) = [x, g] = xgx^{-1}g^{-1}$. This function is sometimes referred to as being the adjoint action. Then this function is nilpotent in the sense that there exists a natural number $n$ such that $f^{n}$, the $n$-th iteration of $f$ sends every element $x$ of $G$ to the identity element.
The group theoretic term nilpotent is simply a transfer from Lie theory: A Lie algebra is nilpotent if $\text{ad}\,x$ is nilpotent for all $x$ (Engel's theorem). Then a connected Lie group is nilpotent if its Lie algebra is nilpotent. Finally, the characterization with upper/lower central series generalized to all groups yielding nilpotent groups in general.