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This paper considers “lazy” random walks supported on a random subset of k elements of a finite group G with order n. If k=⌈a log2 n⌉ where a>1 is constant, then most such walks take no more than a multiple of log2 n steps to get close to uniformly distributed on G. If k=log2 n+f(n) where f(n)→∞ and f(n)/log2 n→0 as n→∞, then most such walks take no more than a multiple of (log2 n) ln(log2 n) steps to get close to uniformly distributed. To get these results, this paper extends techniques of Erdos and Renyi and of Pak.