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 Sn 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 An. 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 Sn, and
of order 3f in An and Sn, 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 An or Sn, given as a black-box group with an order oracle, is
discussed.