1992/11/03 by Jörg Brendle, Brendle, Jörg
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Finite Group Theory Research #Logic (math.LO) #Rings, Modules, and Algebras #math.LO
paper · pdf · doi:10.48550/arxiv.math/9211201
arxiv created 1992/11/03 · openalex publication_date 1992/11/03 · arxiv updated 2016/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a group G, we let Z(G) denote its center, G' its commutator subgroup, and Phi (G) its Frattini subgroup (the intersection of all maximal proper subgroups of G). Given U leq G, we let NG (U) stand for the normalizer of U in G. A group G is FC iff every element g in G has finitely many conjugates. A p-group E is called extraspecial iff Phi (E) = E' = Z(E) cong Zp, the cyclic group with p elements. When generalizing a characterization of centre-by-finite groups due to B. H. Neumann, M. J. Tomkinson asked the following question. Is there an FC-group G with vert G / Z(G) vert = kappa but [G:NG(U)] < kappa for all (abelian) subgroups U of G, where kappa is an uncountable cardinal. We consider this question for kappa = omega1 and kappa = omega2. It turns out that the answer is largely independent of ZFC, and that it differs greatly in the two cases. More explicitly, for kappa = omega1, it is consistent with, and independent of, ZFC that there is a group G with vert G / Z(G) vert = omega1 and [G:NG (A)] leq omega for all abelian A leq G. We do not know whether the same statement is still consistent if we drop abelian. On the other hand, for kappa = omega2, the non-existence of groups G with vert G / Z(G) vert = omega2 and [G : NG (A) ] leq omega1 for all (abelian) A leq G is equiconsistent with the existence of an inaccessible cardinal. In particular, there is an extraspecial p-group with this property if there is a Kurepa tree.