Bicyclic units, Bass cyclic units and free groups

Abstract

Let G be a finite group and ZG its integral group ring. We show that if α is a non-trivial bicyclic unit of ZG, then there are bicyclic units β and γ of different types, such that α, β and α, γ are non-abelian free groups. In case that G is non-abelian of order coprime with 6, then we prove the existence of a bicyclic unit u and a Bass cyclic unit v in ZG, such that for every positive integer m big enough, u m , v is a free non-abelian group. IntroductionA free pair is by definition a pair formed by two generators of a non-abelian free group. Let G be a finite group. The existence of free pairs in the group of units U (ZG) of the integral group ring ZG, was firstly proved by Hartley and Pickel [8], provided that G is neither abelian nor a Hamiltonian 2-group (equivalently U (ZG) is neither abelian nor finite , which prove that the group generated by the bicyclic and the Bass cyclic units generates a big portion of U (ZG), show that these two types of units have an important role in the structure of U (ZG). As a consequence, several authors have payed attention to the problem of describing the structure of the group generated either by bicyclic units, or Bass cyclic units and more specifically, to the problem of deciding when two bicyclic units or Bass cyclic units form a free pair [3,7,10,16]. The bicyclic units of ZG are the elements of one of the following forms

Rational Group Algebras of Finite Groups: From Idempotents to Units of Integral Group Rings

“…Some cases with exceptional components have also been considered, see for example [GS,Jes,JL2,Seh3]). In general, very little is known on the structure of the group generated by the Bass cyclic units and the bicyclic units, except that "often" two of them generate a free group of rank two (see for example [GP,GdR,Jes,JdRR,MS2,JL3]). In this paper, for G a finite nilpotent group, we not only give new generators for a subgroup of finite index, but more importantly, the generating set is divided into three subsets, one of them generating a subgroup of finite index in the central units and each of the other two generates a nilpotent group.…”

Alternating units as free factors in the group of units of integral group rings

“…If w ∈ Z(G), then we are again in the p-critical case, and G is either of type (1), (2) or (3). If G is not a 3-group, then take w ∈ G such that | w | = p is a prime number not equal to 3.…”

“…It is well known [10] that unless G is either an abelian or a Hamiltonian 2-group, U(ZG) always contains free (non-cyclic) subgroups. Other instances of construction of free subgroups of U(ZG), using either Bass cyclic units or bicyclic units, can be seen in [1,3,5,6,8,11]. More recently, this gap was filled by using bicyclic units [12] and by using Bass cyclic units [2].…”

Bass Cyclic Units as Factors in a Free Group in Integral Group Ring Units

“…This goal was reached, using either bicyclic or Bass cyclic units, by combining results of Marciniak and Sehgal [13] and Ferraz [3]. Other constructions of free groups in U (ZG) using either Bass cyclic units or bicyclic units can be found in [2,[6][7][8]12] and [15].…”

“…In this paper we prove a theorem which suggests that this is basically the only obstacle to construct a non-abelian free group in U (ZG) from a given Bass cyclic unit. More precisely in this paper we address the following conjecture which is a natural outgrowth of the result of [6][7][8].…”