From [Wikipedia](https://en.wikipedia.org/wiki/Abelian_group#Historical_remarks):


"**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](https://en.wikipedia.org/wiki/Solvable_group):


"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 ?



Historical reference request on Nilpotent groups
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.


