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?



Origin of the term 'index of a subgroup'
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](//archive.org/details/117768378/page/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](//archive.org/details/117768378/page/n8): Serret’s *Cours d’Algèbre Supérieure* (1866) (“the first connected exposition of the theory”) has, [p. 287](//archive.org/details/coursdalgbresup11serrgoog/page/n305):



> 
> 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](//gallica.bnf.fr/ark:/12148/bpt6k433673r/f8):



> 
> 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](//fr.wikipedia.org/wiki/Sp%C3%A9cial:Index/Indice), especially [ratio](//fr.wikipedia.org/wiki/Indice#Rapport)...), and that the paper’s title itself restates this very definition — it could be shortened “*Sur l'Indice*”.


