Involutions and Free Pairs of Bicyclic Units in Integral Group Rings

If ∗ : G → G is an involution on the finite group G, then ∗ extends to an involution on the integral group ring Z[G]. In this paper, we consider whether bicyclic units u ∈ Z[G] exist with the property that the group 〈u, u∗〉, generated by u and u∗, is free on the two generators. If this occurs, we say that (u, u∗) is a free bicyclic pair. It turns out that the existence of u depends strongly upon the structure of G and on the nature of the involution. One positive result here is that if G is a nonabelian group with all Sylow subgroups abelian, then for any involution ∗, Z[G] contains a free bicyclic pair. 1. Free Pairs of Bicyclic Units Let R be a commutative ring with 1 and let R[G] denote the group ring of G over R. For the most part, we will be concerned with integral group rings where R = Z, or with group algebras where R is a field. Furthermore, we will mainly be interested in finite groups G, although some of our results do hold more generally. If B is a finite subgroup of G, we let B̂ ∈ R[G] denote the sum of the elements of B in R[G]. Since (1− b)B̂ = B̂(1− b) = 0 for any b ∈ B, we see that group ring elements of the form (1− b)aB̂, with a ∈ G, have square 0. Hence 1+(1− b)aB̂ is a unit in the ring R[G] with inverse 1− (1− b)aB̂. When B = 〈b〉 is cyclic, elements of the form u = 1 + (1 − b)aB̂ are known as bicyclic units. It is easy to see that 1+(1−b)aB̂ = 1 if and only if b = a−1ba ∈ B and hence if and only if a ∈ NG(B), the normalizer of B. In particular, if G is a Dedekind group, namely a group with all subgroups normal, then R[G] has no nontrivial bicyclic units. Now suppose that ∗ : G → G is an involution, that is an antiautomorphism of order 2. Then ∗ extends to an involution of R[G]. In particular, if u is a unit of R[G], then so is u∗, and we are interested in the nature of the subgroup 〈u, u∗〉 of the unit group that is generated by these two elements. If ∗ : G→ G is the inverse map and if R[G] = Z[G], then it was shown in [6] that, for every nontrivial bicyclic unit u, the group 〈u, u∗〉 is free of rank 2. Certainly, it is of interest to see whether this property remains true for other involutions, and that is the theme of [2], where groups G with |G : Z(G)| = 4 are considered. In this paper, we study a wider class of examples. For convenience, we say that (u, u∗) is a free bicyclic pair if u is a bicyclic unit with 〈u, u∗〉 a free group on the two generators. Section 1 is devoted to general tools. Following [6], we consider algebras over the complex numbers C and study when units 1+α and 1+β with α = β = 0 generate a free group of rank 2, that is when (1 + α, 1 + β) is a free pair. In some sense, Research supported in part by the grant CNPq 303.756/82-5 and by Fapesp-Brazil, Proj. Tem-