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 Historical Note about abelian group or Commutative

Commutative groups are called abelian in honor of the Norwegian mathematician Niels Henrik Abel (1802 − 1829). Abel was interested in the question of solvability of polynomial equations.

In a paper written in 1828, he proved that if all the roots of such an equation can be expressed as rational functions 𝑓, 𝑔, … , h of one of them, say 𝑥, and if for any two of these roots, 𝑓(𝑥) and 𝑔(𝑥), the relation 𝑓 𝑔 𝑥 = 𝑔 𝑓 𝑥 always holds, then the equation is solvable by radicals. Abel showed that each of these functions in fact permutes the roots of the equation; hence, these functions are elements of the group of permutations of the roots. It was this property of commutativity in these permutation groups associated with solvable equations that led Camille Jordan in his 1870 treatise on algebra to name such groups abeli;the name since then has been applied to commutative group in general.

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