Set-Theoretic Aspects of Periodic FC-Groups - Extraspecial p-Groups and Kurepa Trees

Abstract

When generalizing a characterization of centre-by-finite groups doe to B. H. Neumann, M. J. Tomkinson asked the following question. Is there an FC-group G with | G/ Z( G)| = κ but [ G : N G ( U)] <κ for all (abelian) subgroups U of G, where κ is an uncountable cardinal [16, Question 7A, p. 149]. We consider this question for κ = ω 1 and κ = ω 2. It turns out that the answer is largely independent of ZFC (the usual axioms of set theory), and that it differs greatly in the two cases.