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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"

From Wikipedia's article on solvable groups:

"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 ?
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    $\begingroup$ btw the original French word for solvable was résoluble and has two English translations in use: the most common solvable and also soluble $\endgroup$
    – YCor
    Commented Jun 2, 2016 at 17:38
  • $\begingroup$ Isn't there one which is more British English and the other American English ? $\endgroup$
    – Drike
    Commented Jun 2, 2016 at 17:41
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    $\begingroup$ 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). $\endgroup$
    – Derek Holt
    Commented Jun 2, 2016 at 17:51
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    $\begingroup$ 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. $\endgroup$
    – KConrad
    Commented Jun 2, 2016 at 18:07
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    $\begingroup$ This post seems to be better suited to History of Science and Mathematics, since it mainly historical in nature. $\endgroup$
    – Danu
    Commented Jun 3, 2016 at 12:17

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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.

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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.

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