= Finite intersection of finite-index subgroups
{title2=$[G:H\cap K]\le[G:H][G:K]$}
For <finite-index subgroups> $H,K$ of a <group> $G$, the map of sets of <left cosets>
$$
g(H\cap K)\longmapsto(gH,gK)
$$
is well defined and injective: equality of both components is equivalent to the difference $g'^{-1}g$ belonging to $H\cap K$. Hence the displayed <subgroup index> bound holds, without a <normal subgroup> hypothesis. By <mathematical induction>, the <index of a subgroup> for a finite intersection is at most the product of the individual <subgroup indices>.
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