We investigate the relationship between a finite group and the set of probabilities associated with evaluating words over the group.Given a finite group, a group element and a word, one may consider the probability that a uniformly random evaluation of the associated verbal mapping yields that particular element of the group. For a fixed group, we consider the set of such probabilities obtained by varying over all words and all group elements. It is known that properties of the group are reflected in this associated set of probabilities. For example, if a group is nilpotent then the set of non-zero probabilities associated with that group has a positive lower bound.We seek to further establish the link between a finite group and its set of probabilities. We show how properties of the group, such as nilpotency and verbal subgroup structure are manifested in the properties of its set of probabilities, such as cardinality, the infimum and the corresponding set of accumulation points. We calculate the set of probabilities explicitly for several groups.
|Date of Award||1 Jun 2012|
|Supervisor||Geoffrey Smith (Supervisor)|