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I'm curious as to how groups, rings, fields got their names. Did someone just start calling these structures by those names, or is there a (not entirely) arbitrary reason for them?

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The word "group" is a translation of the French "groupe", which was first used for this purpose by Évariste Galois around 1830.

Galois considered particular permutation groups in the context of permuting the roots of an algebraic equation; his word was applied to the developing concept of an abstract group during the mid-1800s. Arthur Cayley gave the first abstract definition of a finite group in 1854; a fully modern notion of group was given by Walther von Dyck in 1882.

(Wikipedia: History of Group Theory).

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