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Probability and Bias in Generating Supersoluble Groups

Part of: Probabilistic methods in group theory

Published online by Cambridge University Press:  22 December 2015

Eleonora Crestani ,
Giovanni De Franceschi  and
Andrea Lucchini
Eleonora Crestani
Affiliation:
Dipartimento di Matematica, Via Trieste 63, 35121 Padova, Italy (crestani.eleonora@gmail.com; giovanni.defranceschi@auckland.ac.nz; lucchini@math.unipd.it)
Giovanni De Franceschi
Affiliation:
Dipartimento di Matematica, Via Trieste 63, 35121 Padova, Italy (crestani.eleonora@gmail.com; giovanni.defranceschi@auckland.ac.nz; lucchini@math.unipd.it)
Andrea Lucchini
Affiliation:
Dipartimento di Matematica, Via Trieste 63, 35121 Padova, Italy (crestani.eleonora@gmail.com; giovanni.defranceschi@auckland.ac.nz; lucchini@math.unipd.it)
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Abstract

We discuss some questions related to the generation of supersoluble groups. First we prove that the number of elements needed to generate a finite supersoluble group G with good probability can be quite a lot larger than the smallest cardinality d(G) of a generating set of G. Indeed, if G is the free prosupersoluble group of rank d ⩾ 2 and dP(G) is the minimum integer k such that the probability of generating G with k elements is positive, then dP(G) = 2d + 1. In contrast to this, if k – d(G) ⩾ 3, then the distribution of the first component in a k-tuple chosen uniformly in the set of all the k-tuples generating G is not too far from the uniform distribution.

Keywords

supersoluble groupsproduct replacement algorithmgenerationprofinite groups

MSC classification

Secondary: 20P05: Probabilistic methods in group theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 4 , November 2016 , pp. 899 - 909
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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