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.
hmakholm left over Monica
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Excerpt from Jeff Miller's (MacTutor) Earliest Known Uses of Some of the Words of Mathematics
GROUP and GROUP THEORY. Evariste Galois introduced the term group in the form of the expression groupe de l'équation in his "Mémoire sur les conditions de résolubilité des équations par radicaux" (written in 1830 but first published in 1846) Oeuvres mathématiques. p. 417. Cajori (vol. 2, page 83) points out that the modern definition of a group is somewhat different from that of Galois, for whom the term denoted a subgroup of the group of permutations of the roots of a given polynomial.
Group appears in English in Arthur Cayley, "On the theory of groups, as depending on the symbolic equation θn = 1," Philosophical Magazine, 1854, vol. 7, pp. 40-47, Reprinted in Collected Works II, pp. 123-130: "A set of symbols, 1, α, β, ..., all of them different, and such that the product of any two of them (no matter what order), or the product of any one of them into itself, belongs to the set, is said to be a group." The paper also introduced the term theory of groups. At the time this more abstract notion of a group made little impact.
Klein and Lie use the term "closed system" in their "Über diejenigen ebenen Curven, welche durch ein geschlossenes System von einfach unendlich vielen vertauschbaren linearen Transformationen in sich übergehen," Mathematische Annalen, 4, (1871), 50-84. Klein adopted the term *gruppe& in his "Vergleichende Betrachtungen über neuere geometrische Forschungen" written in 1872 and reissued in Mathematische Annalen, 43, (1893),63-100. See the entry ERLANGEN PROGRAM.
Group-theory is found in English in 1888 in George Gavin Morrice's translation of Felix Klein, Lectures on the Ikosahedron and the solution of Equations of the Fifth Degree. [Google print search]
See MacTutor Hilbert-Weyl discussion between 1919 and 1922, occurring e.g. in Weyl's "Über die neue Grundlagenkrise der Mathematik", Math.Z. 10 (1921) 39-79."
Bill Dubuque
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