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Let $Q$ be a probability measure on a finite group $G$, and let $H$ be asubgroup of $G$. We show that a necessary and sufficient condition for therandom walk driven by $Q$ on $G$ to induce a Markov chain on the double cosetspace $H\backslash G/H$, is that $Q(gH)$ is constant as $g$ ranges over anydouble coset of $H$ in $G$. We obtain this result as a corollary of a moregeneral theorem on the double cosets $H \backslash G / K$ for $K$ an arbitrarysubgroup of $G$. As an application we study a variation on the $r$-top torandom shuffle which we show induces an irreducible, recurrent, reversible andergodic Markov chain on the relevant double cosets. The transition matrix ofthe induced walk has remarkable spectral properties: we find its invariantdistribution and its eigenvalues and hence determine its rate of convergence.

A necessary and sufficient condition for double coset lumping of Markov chains on groups with an application to the random to top shuffle