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Let $I$ be a nonempty directed set, and $H_i \le G_i$ both be directed sequence of groups living in some ambient group. Is there a nice formula for $|\bigcup_{i \in I} G_i : \bigcup_{i \in I} H_i|$, perhaps involving a limit or supremum of the $|G_i : H_i|$? I must confess, I don't really have mu...