Groups with the same orders of Sylow normalizers as the Mathieu groups.
Let be a group with the property that there are no infinite descending chains of non-subnormal subgroups of for which all successive indices are infinite. The main result is that if is a locally (soluble-by-finite) group with this property then either has all subgroups subnormal or is a soluble-by-finite minimax group. This result fills a gap left in an earlier paper by the same authors on groups with the stated property.
Hass, Rubinstein, and Scott showed that every closed aspherical (irreducible) 3-manifold whose fundamental group contains the fundamental group of a closed aspherical surface, is covered by Euclidean space. This theorem does not generalize to higher dimensions. However, we provide geometric tools with which variations of this theorem can be proved in all dimensions.
We present a simple constructive proof of the fact that every abelian discrete group is uniformly amenable. We improve the growth function obtained earlier and find the optimal growth function in a particular case. We also compute a growth function for some non-abelian uniformly amenable group.