周期性$FC$-群的集论方面--超特殊p-群和Kurepa树
Set-theoretic aspects of periodic $FC$-groups --- extraspecial p-groups and Kurepa trees
摘要：
给定一个群G，设Z（G）表示它的中心，G '表示它的换位子群，Phi（G）表示它的弗拉蒂尼子群（G的所有极大真子群的交集）.给定U ∈ G，我们用NG（U）表示U在G中的正规化子.群G是FC当且仅当G中的每个元素g都有2个共轭。称p-群E为超特殊群当且仅当Phi（E）= E '= Z（E）cong Z_p是p元的循环群.在推广B. H. Neumann，M. J. Tomkinson提出了以下问题。对于G的所有交换子群U，是否存在一个FC-群G，使得vert G/Z（G）vert = kappa但[G：N_G（U）]<kappa，其中kappa是不可数基数.我们考虑kappa = omega_1和kappa = omega_2的情况。事实证明，答案在很大程度上与ZFC无关，并且在两种情况下差异很大。更明确地说，对于kappa = omega_1，存在一个群G，其vert G/Z（G）vert = omega_1，且对所有交换群Aleq G有[G：N_G（A）] leq omega，这与ZFC是一致的，且与ZFC无关.我们不知道如果去掉阿贝尔，同样的陈述是否仍然是一致的。另一方面，对kappa = omega_2，群G不存在vert G/Z（G）vert = omega_2且[G：N_G（A）] leq omega_1对所有的（交换）Aleq G是与存在一个不可达基数相等一致的.特别地，如果存在Kurepa树，则存在具有此性质的超特殊p-群。
英文摘要：
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.
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