純粋・応用数学（含むガロア理論）6

現代数学の系譜雑談

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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" appears for the first time in 1910 in a paper by G. A. Miller in the Quarterly Journal of Mathmatics (vol. 41, p. 382). Various other terms have been used including the german Nebengruppen (Weber) and conjugate group (Burnside).[11]

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 H of a group of permutations G induced two decompositions of 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 H instead of 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.[6]
Calling the coset gH the left coset of g with respect to H, while most common today,[10] has not been universally true in the past. For instance, Hall (1959) would call gH a right coset, emphasizing the subgroup being on the right.

https://ja.wikipedia.org/wiki/%E5%90%8C%E5%80%A4%E9%96%A2%E4%BF%82
同値関係（どうちかんけい、英: equivalence relation）は反射的、対称的かつ推移的な二項関係を言う。これらの性質の帰結として、与えられた集合において、一つの同値関係はその集合を同値類に分割（類別）する。

同値関係にあることを表す記法は文献によって様々に用いられるけれども、与えられた集合上の同値関係 R に関して二元 a, b が同値であることを "a ～ b" や "a ≡ b" で表すのがもっともよく用いられる記法である。R に関して同値であることを明示する場合には、"a ～R b" や "a ≡R b" あるいは "aRb" などと書かれる。

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