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This paper addresses the fundamental problem of permuting the elements of an array of $n$ elements according to some given permutation. Our goal is to perform the permutation quickly using only a polylogarithmic number of bits of extra storage. The main result is an algorithm whose worst case running time is $O(n \log n)$ and that uses $O(\log n )$ additional $\log n$-bit words of memory. A simpler method is presented for the case in which both the permutation and its inverse can be computed at (amortised) unit cost. This algorithm requires $O(n \log n)$ time and $O(1)$ words in the worst case. These results are extended to the situation in which we are to apply a power of the permutation. A linear time, $O(1)$ word method is presented for the special case in which the data values are all distinct and are either initially in sorted order or will be when permuted.