# Where is the name “coset” in group theory from?

One of the most important application of "coset", I think, is to prove the Lagrange's theorem, which was not originally stated in the group theoretic terms. In some textbooks I have read about abstract algebra, I didn't find any history about "coset".

Here is my question:

Where is the concept "coset" from? And what is it originally used for?

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See Mathword http://jeff560.tripod.com/mathword.html for first use of this.

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...where it says "COSET was used in 1910 by G. A. Miller in Quarterly Journal of Mathematics." –  wildildildlife Jun 25 '11 at 16:24
The first use doesn't directly tell us where the name comes from in the sense of etymology, which is what I think was the request. My guess is that coset might be a contraction of "complementary set", but I have no real evidence for that. –  KCd Jun 25 '11 at 19:38
G. A. Miller is a hero of mine. It is nice that the word traces back to him :) –  Mariano Suárez-Alvarez Jan 20 '12 at 8:38

Gallian's "Contemporary Abstract Algebra" says that Galois invented the concept of a coset in 1830, but the name coset was not used until 1910 by G.A. Miller.

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Did Galois use a word to refer to cosets? –  Mariano Suárez-Alvarez Jan 20 '12 at 8:39
The utility of the concept of cosets extends far beyond Lagrange's Theorem. For (a rather simple) example, in the integers, the coset $1+2\mathbb{Z}$ is the set of all odd integers, which is often very useful as being considered as "a single entity".
In Dutch one uses the same term: nevenklasse. Perhaps one coudl translate it as neighboring class/group or adjacent class/group. Personally I prefer left/right translate instead of left/right coset, especially for topological groups, I find this term more descriptive: $gH$ is what you get when you translate $H$ via $g$. –  wildildildlife Jan 20 '12 at 13:31