A k-Conjugacy Class Problem

• Main Author: Roberts, Collin
• Language: en
• Published: 2007
• Subjects: group theory; k-conjugacy class; locally finite group; universal locally finite group; existentially closed group; Engel group; Pure Mathematics
• Online Access: http://hdl.handle.net/10012/3208

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.