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

R. Beals, C.R. Leedham-Green, Alice Niemeyer, Cheryl Praeger, Akos Seress

Research output: Contribution to journalArticle

21 Citations (Scopus)

### 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 ξ Sn is an element of S-n is an n-cycle, given that σn = 1, is at least 1/10.
Original language English 2097-2113 Transactions of the American Mathematical Society 355 5 https://doi.org/10.1090/S0002-9947-03-03040-X Published - 2003

### Fingerprint

Alternating group
Black Box
Symmetric group
Permutation Representation
Finite Group
Random Element
Matrix Groups
Permutation group
Conditional probability
Isomorphism
Isomorphic
Cycle
Standards

### Cite this

Beals, R. ; Leedham-Green, C.R. ; Niemeyer, Alice ; Praeger, Cheryl ; Seress, Akos. / A Black-Box Group Algorithm for Recognizing Finite Symmetric and Alternating Groups, I. In: Transactions of the American Mathematical Society. 2003 ; Vol. 355, No. 5. pp. 2097-2113.
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A Black-Box Group Algorithm for Recognizing Finite Symmetric and Alternating Groups, I. / Beals, R.; Leedham-Green, C.R.; Niemeyer, Alice; Praeger, Cheryl; Seress, Akos.

In: Transactions of the American Mathematical Society, Vol. 355, No. 5, 2003, p. 2097-2113.

Research output: Contribution to journalArticle

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

AU - Beals, R.

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AU - Seress, Akos

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