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. [ABSTRACT FROM AUTHOR]