Integral group rings with all central units trivial

Abstract

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.

Finite simple groups with short Galois orbits on conjugacy classes

“…This group contains a cyclic subgroup of order divisible by 1 d (q (n−2)/2 + 1). We have ϕ 1 d (q (n−2)/2 + 1) > 8(n − 2) by the proof of Lemma 5.4, where d = (2, q + 1) unless (n, q) ∈ {(8, 2), (8,3), (8,4), (8,5), (10,2), (10,3), (12,2), (12,3), (14,2), (16,2)}.…”

“…We give a generalization of this statement in Proposition 2.2. As a natural consequence of A. Bovdi's result, a not necessarily finite group G is called a cut-group if all central units of ZG are trivial (of the form ±g where g ∈ Z(G)), see [10]. The first results on cut-groups were obtained by A.…”

“…The first results on cut-groups were obtained by A. Bovdi and Patay. Recent results and references on this topic can be found in [8,10] and [11].…”

Integral Group Rings With All Central Units Trivial: Solvable Groups

“…Observe that in either of the two possibilities given by (2), for the value of o(x), 5 is a primitive root of o(x). Therefore, (4) yields that ϕ(o(x)) divides o(g), where ϕ denotes the Euler's phi function.…”

Group rings and the RS property

“…Given a group G, let U(Z[G]) be the group of units of the integral group ring Z[G] and let Z(U(Z[G])) be its center. Trivially, Z(U(Z[G])) contains ±Z(G), where Z(G) denotes the center of G. In case Z(U(Z[G])) = ±Z(G), i.e., all central units of Z[G] are trivial, following [3], we call G a cut-group or a group with the cut-property. The question of classifying cut-groups was explicitly posed, for the first time, by Goodaire and Parmenter [8].…”