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Let  G  be a finite group and ℤ G  its integral group ring. We show that if α is a nontrivial bicyclic unit of ℤ G , then there are bicyclic units β and γ of different types, such that 〈α, β〉 and 〈α, γ〉 are non-abelian free groups. In the case when  G  is non-abelian of order coprime to 6 we prove the existence of a bicyclic unit  u  and a Bass cyclic unit  v  in ℤ G , such that 〈 u m  ,  v 〉 is a free non-abelian group for all sufficiently large positive integers  m .

Bicyclic units, Bass cyclic units and free groups