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The index of a subgroup $H$ in a group $G$ is the number of distinct cosets of $H$ in $G$.

Why did someone decide to call this an 'index'? What's the rationale for this?

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The short answer is Cauchy, with only justification: “for short”. Burnside’s Theory of groups of finite order (1897) has a useful glossary stating, p. 382 (my bold):

The ratio of the order of a sub-group $H$, to the order of the group $G$ containing it, is called by French writers the “indice,” by German the “Index,” of $H$ in $G$. The phrase is most commonly used of a substitution group in relation to the symmetric group of the same degree.

The French writers in question are those mentioned in Burnside’s preface: Serret’s Cours d’Algèbre Supérieure (1866) (“the first connected exposition of the theory”) has, p. 287:

432. $\vphantom{\frac{\mathrm N}m}$ To abbreviate the discourse, I will call index of a system of conjugate substitutions formed with $n$ letters, the quotient obtained by dividing the product $\mathrm N=1.2.3\dots n$ by the number which expresses the system’s order. If $m$ denotes the index of a conjugate system of order $\mu$, one will have $m=\frac{\mathrm N}\mu$.

And already Cauchy’s memoir Sur le Nombre de Valeurs qu’une Fonction peut acquérir, lorsqu’on y permute de toutes les manières possibles les quantités qu’elle renferme, J. École Polytechnique 10 (1815) 1–28 (from which “the theory of groups of finite order may be said to date”) has, p. 6:

if one represents by $R$ the total number of essentially different values of the function $K$, $M$ being the number of values equivalent to $K_\alpha$ (...) one will have $RM=N$ (...) Hence $R$ (...) can only be a factor of $N$, that is, of the product $1.2.3\dots n$ (...) For short, I will henceforth call index of the function $K$, the number $R$ which indicates how many essentially different values this function can assume.

Note that the index is to “indicate” (but this could be said of any named count, especially ratio...), and that the paper’s title itself restates this very definition — it could be shortened “Sur l'Indice”.

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    $\begingroup$ Let me add that "indice" is used in French in the meaning "subscript". $\endgroup$
    – YCor
    May 8 '20 at 21:26

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