TY - JOUR

T1 - A Black-Box Group Algorithm for Recognizing Finite Symmetric and Alternating Groups, I

AU - Beals, R.

AU - Leedham-Green, C.R.

AU - Niemeyer, Alice

AU - Praeger, Cheryl

AU - Seress, Akos

PY - 2003

Y1 - 2003

N2 - 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 σ ξ Sn is an element of S-n is an n-cycle, given that σn = 1, is at least 1/10.

AB - 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 σ ξ Sn is an element of S-n is an n-cycle, given that σn = 1, is at least 1/10.

U2 - 10.1090/S0002-9947-03-03040-X

DO - 10.1090/S0002-9947-03-03040-X

M3 - Article

VL - 355

SP - 2097

EP - 2113

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -