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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8See this question and this question on math.SE, and this question on MathOverflow (and this question and this question for other terminology). – Zev Chonoles Jul 02 '12 at 00:27
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1See http://en.wikipedia.org/wiki/Group_(mathematics)#History – lhf Jul 02 '12 at 00:34
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2I think Galois used the word group to refer to permutation groups of zeros of polynomials. I think Arthur Cayley is the author of the modern definition, but he was probably just using established terminology rather than endorsing the idea that that's the best possible word. – Michael Hardy Jul 02 '12 at 01:13
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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.
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