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.
Probability and Bias in Generating Supersoluble Groups
CRESTANI, ELEONORA;LUCCHINI, ANDREA
2016
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.Pubblicazioni consigliate
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