Springer Science and Business Media LLC

A group G is said to be a modular group if it has modular subgroup lattice. We will prove in this paper that a polycyclic group G is modular if and only if all its finite homomorphic images are modular groups. Similar results will also be obtained for other conditions of modular type.

In a real normed space $ (E, \| \cdot \|) $ we consider the perpendicular bisectors associated to any linear segment, and using them we give some characterizations of inner product spaces. Based on the properties of the perpendicular bisectors in the real plane, we introduce a new orthogonal relation in $ (E, \| \cdot \|) $ and its properties are investigated.

Let $$Q_8$$ be the quaternion group of order 8 and $${\chi }$$ its faithful irreducible character. Then $${\chi }$$ can be realized over certain imaginary quadratic number fields $$K={\mathbb Q}\bigl (\sqrt{-N}\bigr )$$ but not over their rings of integers (Feit, Serre); here N is a positive square-free integer. We show that this happens precisely when $${\mathbb Q}\bigl (\sqrt{N}\bigr )$$ but not $${\mathbb Q}\bigl (\sqrt{2}, \sqrt{N}\bigr )$$ can be embedded into a $$Q_8$$ -field over the rationals (Galois with group $$Q_8$$ ) and N is not a sum of two integer squares. In particular, we get that $${\chi }$$ cannot be integrally realized if N is (properly) divisible by some prime $$q\equiv 7\,({\textrm{mod}\,}8)$$ .

We provide a new geometric proof of Reimann’s theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in Heisenberg groups as well as in more general stratified nilpotent Carnot groups.