Parker Vectors for Infinite Groups

Parker Vectors for Infinite Groups

We take the first step towards establishing a theory of Parker vectors for infinite permutation groups, with an emphasis towards oligomorphic groups. We show that, on the one hand, many results for finite groups extend naturally to the infinite case (Parker’s Lemma, multiplicative properties, etc.), while on the other, in the infinite case some genuinely new phenomena arise. We also note that c...

Sequences realized as Parker vectors of oligomorphic permutation groups

The purpose of this paper is to study the Parker vectors (in fact, sequences) of several known classes of oligomorphic groups. The Parker sequence of a group G is the sequence that counts the number of G-orbits on cycles appearing in elements of G. This work was inspired by Cameron’s paper on the sequences realized by counting orbits on k-sets and k-tuples.

Infinite Groups

Most interesting groups arise as a group of transformations. For example the set of “rigid motions” of 2-dimensional space R forms a group denoted Isom(R). There are three kinds of transformations: reflection about a line, rotation about a point, and translation in a direction. These form a group under composition (i.e., do one transformation and then do the other). For example, the composition...

Universal for Infinite Symmetric Groups ?

Let W= W(xx,...,Xj)bc any word in thej free generators xx,...,Xj, and suppose that W cannot be expressed in the form W = Vk for V a word and for k an integer with \k\¥" 1. We ask whether the equation f=W has a solution (xx>... ,Xj) = (a,,... ,Oj) €E GJ whenever G is an infinite symmetric group and/is an element in G. We establish an affirmative answer in the case that W(x, y) = xmy" for m and n...

Infinite Permutation Groups