Construction of central units in integral group rings of finite groups

Abstract

Abstract. In this paper we give new constructions of central units that generate a subgroup of finite index in the central units of the integral group ring ZG of a finite group. This is done for a very large class of finite groups G, including the abelian-by-supersolvable groups.Let G be a finite group. When G is abelian it is well known (and due to Bass and Milnor; see [1] and [9, Theorem 12.7 and Theorem 13.1]) that the Bass cyclic units of the integral group ring ZG generate a subgroup of finite index in Z(U(ZG)), the group of (central) units of ZG. In [4], it was shown that if G is a nilpotent finite group of nilpotency class n, then the group b (n) is of finite index in Z(U(ZG)).Here b (n) is defined recursively as follows. We denote by Z i the i-th centre of G. For any x ∈ G and Bass cyclic unit b ∈ Z x , put b (1) = b, and, for 2 ≤ i ≤ n, putRecently, Ferraz and Simón [2] gave a construction of central units in case G is meta-(cyclic of prime order) that generate a subgroup of finite index. They actually provided an independent set of generators.In this note we continue these investigations and provide new constructions of central units that generate a subgroup of finite index in the central units of ZG and this for a very large class of finite groups G, including the abelian-by-supersolvable groups. The proof and construction relies on a beautiful paper of Olivieri, del Río and Simón [6] in which they give an explicit construction of the primitive central idempotents of rational group algebras QG of the so-called strongly monomial groups (for example, abelian-by-supersolvable groups). Our proof thus uses a completely different method from the one used in [4] for finite nilpotent groups. In Section 1 we will recall the necessary background on this. In Section 2 we prove the main results for strongly monomial groups. In Section 3 we prove a general result that yields generators for central units from subgroups and quotient groups. An application is given for solvable Frobenius groups.

Central units of integral group rings

“…In this section we construct generalized Bass units and show that the group they generate contains a subgroup of finite index in the central units of the integral group ring ZG for finite strongly monomial groups G. This generalizes Corollary 2.3 in [JP12] on generators for central units of the integral group ring of a finite metabelian group.…”

“…The idea originates from [JPS96], in which the authors constructed central units in ZG based on Bass units b ∈ ZG for finite nilpotent groups G. We denote by Z i the i-th center, i.e. Z 0 = 1 and…”

“…For finite nilpotent groups G, constructions of central units of this type have earlier been considered (using K-theory) by Jespers, Parmenter, Sehgal [JPS96] in the context of finding finitely many generators for a subgroup of finite index in the center of U(ZG). Ferraz and Simón in [FS08] constructed a basis for the center of U(ZG) in case G is a metacyclic group of order pq, with p and q two distinct odd primes.…”

Integral group rings with all central units trivial

“…The structure of Z(U(Z[G])) has been a * Research supported by IISER, Mohali, India, is gratefully acknowledged † Corresponding author subject of intensive research (see e.g. [4,10,11,14,15,16,17,18,19,25]). Clearly Z(U(Z[G])) contains the so-called trivial central units ±g, g ∈ Z(G), the centre of G. Thus there arises the problem of characterizing the groups G having the property that all central units of the integral group ring Z[G] are trivial.…”

CHARACTERIZATION OF CENTRAL UNITS OF ℤAn

“…On the other hand, Jespers, Parmenter, and Sehgal [6] found a different set of generators which works for finitely generated nilpotent groups and constructed these generators from Bass cyclic units in ZG. The construction depended on the existence of a finite normal series in G. If we choose A n (n > 4) as a finite group, it is impossible to construct generators for U(Z(ZG)) from Bass cyclic units since A n is a simple group.…”