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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. 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.
Received: 2008-04-01
Revised: 2009-09-26
Published Online: 2010-04-23
Published in Print: 2010-September
© de Gruyter 2010
