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Estimation Problems and Randomised Group Algorithms

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Book cover Probabilistic Group Theory, Combinatorics, and Computing

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2070))

Abstract

This chapter discusses the role of estimation in the design and analysis of randomised algorithms for computing with finite groups.An exposition is given of a variety of different approaches to estimating proportions of important element classes, including geometric methods, and the use of generating functions and the theory of Lie type groups.Numerous results concerning estimation in permutation groups and finite classical groups are surveyed.An application is given to the construction of involution centralisers, a crucial component in the constructive recognition of finite simple groups.

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Notes

  1. 1.

    Translation by Peter M. Neumann, The Queen’s College, University of Oxford.

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Acknowledgements

This chapter forms part of our Australian Research Council Discovery Project DP110101153. Praeger and Seress are supported by an Australian Research Council Federation Fellowship and Professorial Fellowship, respectively. Niemeyer thanks the Lehrstuhl D für Mathematik at RWTH Aachen for their hospitality, and acknowledges a DFG grant in SPP1489. All three of us warmly thank the de Brún Centre for Computational Algebra at National University of Ireland, Galway, for their hospitality during the Workshop on Groups, Combinatorics and Computing in April 2011, where we presented the short lecture course that led to the development of this chapter. We are very grateful to Peter M. Neumann for many thoughtful comments and advice, and his translation of Euler’s words in Sect. 2.2.2.

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Authors and Affiliations

  1. Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Alice C. Niemeyer, Cheryl E. Praeger & Ákos Seress

  2. King Abdulaziz University, Jeddah, Saudi Arabia

    Cheryl E. Praeger

  3. The Ohio State University, Columbus, OH, USA

    Ákos Seress

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  1. Alice C. Niemeyer
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Correspondence to Alice C. Niemeyer .

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  1. , Department of Mathematics, National University of Ireland, University Road, Galway, 0, Ireland

    Alla Detinko

  2. , Department of Mathematics, National University of Ireland, University Road, Galway, Ireland

    Dane Flannery

  3. , Mathematics, University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    Eamonn O'Brien

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Niemeyer, A.C., Praeger, C.E., Seress, Á. (2013). Estimation Problems and Randomised Group Algorithms. In: Detinko, A., Flannery, D., O'Brien, E. (eds) Probabilistic Group Theory, Combinatorics, and Computing. Lecture Notes in Mathematics, vol 2070. Springer, London. https://doi.org/10.1007/978-1-4471-4814-2_2

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