The Cyclizer Series of Infinite Permutation Groups

Abstract

The cyclizer of an innite permutation group G is the group generated by thecycles involved in elements of G, along with G itself. There is an ascending subgroupseries beginning with G, where each term in the series is the cyclizer ofthe previous term. We call this series the cyclizer series for G. If this seriesterminates then we say the cyclizer length of G is the length of the respectivecyclizer series. We study several innite permutation groups, and either determinetheir cyclizer series, or determine that the cyclizer series terminates andgive the cyclizer length. In each of the innite permutation groups studied, thecyclizer length is at most 3. We also study the structure of a group that arisesas the cyclizer of the innite cyclic group acting regularly on itself. Our studydiscovers an interesting innite simple group, and a family of associated innitecharacteristically simple groups.

Date of Award	22 May 2013
Original language	English
Awarding Institution	University of Bath
Supervisor	Geoffrey Smith (Supervisor)