## A black-box group algorithm for recognizing finite symmetric and alternating groups, I

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- by Robert Beals, Charles R. Leedham-Green, Alice C. Niemeyer, Cheryl E. Praeger and Ákos Seress PDF
- Trans. Amer. Math. Soc.
**355**(2003), 2097-2113 Request permission

## 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 $\sigma \in S_n$ is an $n$-cycle, given that $\sigma ^n=1$, is at least $1/10$.## References

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## Additional Information

**Robert Beals**- Affiliation: IDA Center for Communications Research, 805 Bunn Drive, Princeton, New Jersey 08540, USA
- Email: beals@idaccr.org
**Charles R. Leedham-Green**- Affiliation: School of Mathematical Sciences, Queen Mary and Westfield College, London E1 4NS, United Kingdom
- Email: crlg@maths.qmw.ac.uk
**Alice C. Niemeyer**- Affiliation: Department of Mathematics and Statistics, University of Western Australia, Crawley, Western Australia 6009, Australia
- Email: alice@maths.uwa.edu.au
**Cheryl E. Praeger**- Affiliation: Department of Mathematics and Statistics, University of Western Australia, Crawley, Western Australia 6009, Australia
- MR Author ID: 141715
- ORCID: 0000-0002-0881-7336
- Email: praeger@maths.uwa.edu.au
**Ákos Seress**- Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA
- Email: akos@math.ohio-state.edu.
- Received by editor(s): September 1, 2000
- Received by editor(s) in revised form: February 18, 2002
- Published electronically: January 14, 2003
- Additional Notes: The third and fourth authors are partially supported by an Australian Research Council Large Grant

The fifth author is partially supported by the National Science Foundation - © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**355**(2003), 2097-2113 - MSC (2000): Primary 20P05, 20C40; Secondary 20B30, 68Q25
- DOI: https://doi.org/10.1090/S0002-9947-03-03040-X
- MathSciNet review: 1953539