The object of study in this paper is the finite groups whose integral group rings have only trivial central units. Prime-power groups and metacyclic groups with this property are characterized. Metacyclic groups are classified according to the central height of the unit groups of their integral group rings.

We provide algorithms to compute a complete irredundant set of extremely strong Shoda pairs of a finite group G and the set of primitive central idempotents of the rational group algebra Q[G] realized by them. These algorithms are also extended to write new algorithms for computing a complete irredundant set of strong Shoda pairs of G and the set of primitive central idempotents of Q[G] realized by them. Another algorithm to check whether a finite group G is normally monomial or not is also described.

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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.

The object of this paper is to examine finite solvable groups whose integral group rings have only trivial central units.