Probability and Bias in Generating Supersoluble Groups

Abstract

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