A Black-box Group Algorithm for Recognizing Finite Symmetric and Alternating Groups, I
نویسندگان
چکیده
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 Sn: the conditional probability that a random element σ ∈ Sn is an n-cycle, given that σn = 1, is at least 1/10.
منابع مشابه
On the Frequency of Permutations Containing a Long Cycle
A general explicit upper bound is obtained for the proportion P (n, m) of elements of order dividing m, where n−1 ≤ m ≤ cn for some constant c, in the finite symmetric group Sn. This is used to find lower bounds for the conditional probabilities that an element of Sn or An contains an r-cycle, given that it satisfies an equation of the form xrs = 1 where s ≤ 3. For example, the conditional prob...
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