The cyclizer series of infinite permutation groups

• Main Author: Turner, Simon
• Other Authors: Smith, Geoffrey
• Published: University of Bath 2013
• Subjects: 512.21
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.577751

The cyclizer of an infinite permutation group G is the group generated by the cycles involved in elements of G, along with G itself. There is an ascending subgroup series beginning with G, where each term in the series is the cyclizer of the previous term. We call this series the cyclizer series for G. If this series terminates then we say the cyclizer length of G is the length of the respective cyclizer series. We study several innite permutation groups, and either determine their cyclizer series, or determine that the cyclizer series terminates and give the cyclizer length. In each of the innite permutation groups studied, the cyclizer length is at most 3. We also study the structure of a group that arises as the cyclizer of the innite cyclic group acting regularly on itself. Our study discovers an interesting innite simple group, and a family of associated innite characteristically simple groups.