On the Probability That a Group Element is p-Singular

Suppose that G is a permutation group of degree n and let p be a prime divisor of /G/. In computational group theory it is a natural and important problem to find an element of G of order p. A polynomial-time (but impractical) algorithm for this is given in [Ka). In practice, an element of the desired type is obtained by "randomly" choosing elements of G and computing their orders. After a few tries, and with some luck, a p-singular element (i.e., one of order divisible by p) frequently turns up. The purpose of this article is to make it clear just how well this procedure can be expected to work. MAIN THEOREM. Let G be a pennutation group of degree n whose order is dil'isible by a prime p. The following then hold. (a) The probability that an element of G has order divisible by p is at least lin. (b) Equality occurs above if and only if either G is sharply 2-transitil'e with fl a power of p or G is the full symmetric group Sn with fl = P ~ 5. It is easy to see in the two situations described in (b) that the probability that a random element is p-singular is exactly l/n. The proof of the rest

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