Editing Coset (section)

== History ==
The concept of a coset dates back to [[Galois]]'s work of 1830–31. He introduced a notation but did not provide a name for the concept. The term "co-set" apparently appears for the first time in 1910 in a paper by G. A. Miller in the ''[[Quarterly Journal of Pure and Applied Mathematics]]'' (vol. 41, p.&nbsp;382). Various other terms have been used including the German ''Nebengruppen'' ([[Eduard Ritter von Weber|Weber]]) and ''conjugate group'' ([[William Burnside|Burnside]]).<ref>{{harvnb|Miller|2012|loc=p. 24 footnote}}</ref> (Note that Miller abbreviated his self-citation to the ''Quarterly Journal of Mathematics''; this does not refer to the [[Quarterly Journal of Mathematics|journal of the same name]], which did not start publication until 1930.)

Galois was concerned with deciding when a given [[polynomial equation]] was [[solvable by radicals]]. A tool that he developed was in noting that a subgroup {{mvar|H}} of a group of [[permutation]]s {{mvar|G}} induced two decompositions of {{mvar|G}} (what we now call left and right cosets). If these decompositions coincided, that is, if the left cosets are the same as the right cosets, then there was a way to reduce the problem to one of working over {{mvar|H}} instead of {{mvar|G}}. [[Camille Jordan]] in his commentaries on Galois's work in 1865 and 1869 elaborated on these ideas and defined normal subgroups as we have above, although he did not use this term.<ref name=Fraleigh />

Calling the coset {{mvar|gH}} the ''left coset'' of {{mvar|g}} with respect to {{mvar|H}}, while most common today,<ref name=Jacobson /> has not been universally true in the past. For instance, {{harvtxt|Hall|1959}} would call {{mvar|gH}} a ''right coset'', emphasizing the subgroup being on the right.