Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups

Let [Formula: see text] be an extension of a finite characteristically simple group by an abelian group or a finite simple group. It is shown that every Coleman automorphism of [Formula: see text] is an inner automorphism. Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.

Global and local properties of finite groups with only finitely many central units in their integral group ring

The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group, we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same; in particular, the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.