Fibres of words in finite groups : a probabilistic approach

We investigate the relationship between a nite group and the set of probabilities associated with evaluating words over the group. Given a nite 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 xed 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 re ected 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 nite 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 inmum and the corresponding set of accumulation points. We calculate the set of probabilities explicitly for several groups. ii