A k-Conjugacy Class Problem

group theory;k-conjugacy class;locally finite group;universal locally finite group;existentially closed group;Engel group;Pure Mathematics

Roberts, Collin

University of Waterloo

关键词: group theory;    k-conjugacy class;    locally finite group;    universal locally finite group;    existentially closed group;    Engel group;    Pure Mathematics;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/3208/1/CollinRobertsMMathThesis.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k.When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc.In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that:(G has finitely many k-conjugacy classes) implies (G is finite)?Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have.We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.