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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 are my questions:

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

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.. and what are they called in other languages ... – GEdgar Jun 2 '14 at 17:15
up vote 3 down vote accepted

See Mathword 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

The answer is relatively simple. It does appear that G.A. Miller did indeed originate the use of the term in his early publications on group theory. It literally means: "co-set". The prefix co- is from the Latin "com-" meaning (among other things) "together with" (as an example, the Spanish derivative "con" simply means "with"). A "co-set" of a subgroup H of a group G, is a subset of G "occuring with" H, and sharing the most important property of H as a set, which is its cardinality. It is perfectly natural to think of the cosets gH of a subgroup H as "sister sets" of the subgroup H, each one formed by multiplying H by some element g of the parent group G. This meaning of co- is the same one we use in such words as co-pilot, or co-worker.

I think it doubtful that "co-" was intended as an abbreviation for "complementary" or "common", as the modern penchant for abbreviation was not so common at the start of the 20th century.

In any case, in Theory and Applications of Finite Groups (1916), Miller indicates that he originated the term, the previous one being Nebengruppen, which I believe translates roughly as "subgroup". Perhaps a better term might have been "translate" of H, a term used in the study of topological groups (esp. Lie groups). It appears that the original use of the word "co-set" was its current use.

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".

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Perhaps somebody who is native German speaker will give a better explanation, but I think that a reasonable translation of Nebengruppe could be something like close group, neighborhood group, since the preposition neben means near, next to. Based on this article from German Wikipedia, it seems that the term Nebenklassen is used today. – Martin Sleziak Jan 20 '12 at 9:01
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
@wildildildlife: the term for coset in Russian is also "adjacent class". – KCd Jun 4 '14 at 13:05

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

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