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**roninpro** For problem 2, we can assume that $\displaystyle |G:K|$ and $\displaystyle |K:H|$ are finite. Let the collection of left cosets of $\displaystyle K$ in $\displaystyle G$ be $\displaystyle \{a_iK\ |\ i=1,2,\ldots r\}$ and the left cosets of $\displaystyle H$ in $\displaystyle K$ be $\displaystyle \{b_jH\ |\ j=1,2,\ldots s\}$. It would suffice to show that the set $\displaystyle \{a_ib_jH\ |\ i=1,2,\ldots r; j=1,2,\ldots s\}$ is the collection of left cosets of $\displaystyle H$ in $\displaystyle G$.