We show that the basic problems of permutation group manipulation admit eecient parallel solutions. Given a permutation group G by a list of generators, we nd a set of NC-eecient strong generators in NC. Using this, we show, that the following problems are in NC: membership in G; determining the order of G; nding the center of G; nding a composition series of G along with permutation representa...

We show that non-trivial extremely amenable topological groups are essentially the same thing as permutation models of the Boolean prime ideal theorem that do not satisfy the axiom of choice. Both are described in terms of partition properties of group actions.

A finite permutation group is cycle-closed if it contains all the cycles of all of its elements. It is shown by elementary means that the cycle-closed groups are precisely the direct products of symmetric groups and cyclic groups of prime order. Moreover, from any group, a cycle-closed group is reached in at most three steps, a step consisting of adding all cycles of all group elements. For inf...

Given a set S with n elements, consider all the possible one-to-one and onto functions from S to itself. This collection of functions is called the permutation group of S, because the functions are simply permuting the elements of S. We notice immediately that it doesn’t matter what the elements of S are (numbers, planets, tacos, etc) just that there are n distinct ones in the set, so we may re...