Set-theoretic aspects of periodic FCFC-groups --- extraspecial p-groups and Kurepa trees

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 N_G (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 Z_p, 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:N_G(U)] < kappa for all (abelian) subgroups U of G, where kappa is an uncountable cardinal. We consider this question for kappa = omega_1 and kappa = omega_2. It turns out that the answer is largely independent of ZFC, and that it differs greatly in the two cases. More explicitly, for kappa = omega_1, it is consistent with, and independent of, ZFC that there is a group G with vert G / Z(G) vert = omega_1 and [G:N_G (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 = omega_2, the non-existence of groups G with vert G / Z(G) vert = omega_2 and [G : N_G (A) ] leq omega_1 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

Strolling through Paradise

With each of the usual tree forcings I (e.g., I = Sacks forcing S, Laver forcing L, Miller forcing M, Mathias forcing R, etc.) we associate a sigma--ideal i^0 on the reals as follows: A \in i^0 iff for all T \in I there is S \leq T (i.e. S is stronger than T or, equivalently, S is a subtree of T) such that A \cap [S] = \emptyset, where [S] denotes the set of branches through S. So, s^0 is the ideal of Marczewski null sets, r^0 is the ideal of Ramsey null sets (nowhere Ramsey sets) etc. We show (in ZFC) that whenever i^0, j^0 are two such ideals, then i^0 \sem j^0 \neq \emptyset. E.g., for I=S and J=R this gives a Marczewski null set which is not Ramsey, extending earlier partial results by Aniszczyk, Frankiewicz, Plewik, Brown and Corazza and answering a question of the latter. In case I=M and J=L this gives a Miller null set which is not Laver null; this answers a question addressed by Spinas. We also investigate the question which pairs of the ideals considered are orthogonal and which are not. Furthermore we include Mycielski's ideal P_2 in our discussion