Abstract
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. The main result here is that if G is a non-nilpotent group, then for any involution, Z[G] contains a free bicyclic pair.
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