## Abstract

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 ∈ S_{n} is an n-cycle, given that σ^{n} = 1, is at least 1/10.

Original language | English (US) |
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Pages (from-to) | 2097-2113 |

Number of pages | 17 |

Journal | Transactions of the American Mathematical Society |

Volume | 355 |

Issue number | 5 |

DOIs | |

State | Published - May 2003 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

## Keywords

- Black-box groups
- Constructive recognition
- Probabilistic method
- Symmetric groups