From <a href="https://en.wikipedia.org/wiki/Abelian_group#Historical_remarks">Wikipedia</a>:

"<b>Abelian groups</b> 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 <a href="https://en.wikipedia.org/wiki/Solvable_group">Wikipedia's article on solvable groups</a>:

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

<blockquote><b>Question.</b> Where do <b>nilpotent groups</b> come from ? Who defined them first ? For what purpose ? Any reference ?</blockquote>