Bicyclic units, Bass cyclic units and free groups

Involutions and Free Pairs of Bicyclic Units in Integral Group Rings of Non-nilpotent Groups

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 upo...

Involutions and Free Pairs of Bass Cyclic Units in Integral Group Rings

Let ZG be the integral group ring of the finite nonabelian group G over the ring of integers Z, and let ∗ be an involution of ZG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (uk,m(x), uk,m(x ∗)), or (uk,m(x), uk,m(y)), formed respectively by Bass cyclic and ∗-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ZG. 1....

Groups generated by two bicyclic units in integral group rings

Bass Units as Free Factors in Integral Group Rings of Simple Groups

Let G be a finite group, u a Bass unit based on an element a of G of prime order, and assume that u has infinite order modulo the center of the units of the integral group ring ZG. It was recently proved that if G is solvable then there is a Bass unit or a bicyclic unit v and a positive integer n such that the group generated by un and vn is a non-abelian free group. It has been conjectured tha...

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 upo...