| Summary: | 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.
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