We present a "Practical" algorithm to construct random elements of a finite group. We analyse its theoretical behaviour and prove that asymptotically it produces uniformly distributed tuples of elements. We discuss tests to assess its effectiveness and use these to decide when its results are acceptable for some matrix groups.

We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups. In this paper, we handle the case when the degree $n$ of the standard permutation representation is part of the input. In a sequel, we shall treat the case when the value of $n$ is not known in advance. As an important ingredient in the theoretical basis for the algorithm, we prove the following result about the orders of elements of $S_n$ : the conditional probability that a random element $\sigma \in S_n$ is an $n$ -cycle, given that $\sigma ^n=1$ , is at least $1/10$ .

Let q be an integer with q [ges ] 2. We give a new proof of a result of Erdös and Turán determining the proportion of elements of the finite symmetric group S_n having no cycle of length a multiple of q. We then extend our methods to the more difficult case of obtaining the proportion of such elements in the finite alternating group A_n. In both cases, we derive an asymptotic formula with error term for the above mentioned proportion, which contains an unexpected occurrence of the Gamma-function.We apply these results to estimate the proportion of elements of order 2f in S_n, and of order 3f in A_n and S_n, where gcd(2, f) = 1, and gcd(3, f) = 1, respectively, and log f is polylogarithmic in n. We also give estimates for the probability that the fth power of such elements is a transposition or a 3-cycle, respectively. An algorithmic application of these results to computing in A_n or S_n, given as a black-box group with an order oracle, is discussed.

The length of every pair {A, B} of n×n complex matrices is at most 2n − 2, if n ≤ 5. That is, for n ≤ 5, the (possibly empty) words in A, B of length at most 2n − 2 span the unital algebra  generated by A, B. For every positive integer m there exist m×m complex matrices C, D such that the length of the pair {C, D} is 2m – 2.