Cycle Types in Infinite Permutation Groups

Infinite Permutation Groups

Infinite Bounded Permutation Groups

This work concerns permutation groups on uncountable sets. The environment is ZFC, Set Theory with Axiom of Choice. Therefore the cardinal numbers are wellordered, a cardinal number k is infinite if and only if Ko < k, and each cardinal number k has a unique successor k. Throughout the paper Cl will be a set (usually infinite), n its cardinality, and m a cardinal number less than n. The support...

Aspects of infinite permutation groups

Until 1980, there was no such subgroup as ‘infinite permutation groups’, according to the Mathematics Subject Classification: permutation groups were assumed to be finite. There were a few papers, for example [10, 62], and a set of lecture notes by Wielandt [72], from the 1950s. Now, however, there are far more papers on the topic than can possibly be summarised in an article like this one. I s...

Cycle-Closed Permutation Groups

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...

Suborbits in Infinite Primitive Permutation Groups

For every infinite cardinal κ, we construct a primitive permutation group which has a finite suborbit paired with a suborbit of size κ. This answers a question of Peter M. Neumann.