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 ?