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I know that if $H$ and $K$ are subgroups of a finite group $G$, then $|HK|=\frac{|H||K|}{|H\cap K|}$.

Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a finite group $G$?

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    There is a lower bound, namely $\frac{|H||K|}{|H\cap K|}$. From page 41, (Hungerford's book Algebra) $[H \vee K: H]\geq [K:H\cap K]$.2012-01-31

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