On free products inside the unit group of integral group rings

Abstract

Abstract We use bicyclic units to give an explicit construction of a subgroup of isomorphic to the free product of two free abelian groups of rank two, assuming that G is a finite nilpotent group and it contains an element g of odd prime order such that the subgroup is not normal in G. To do this we first construct a subgroup isomorphic to the desired free product inside and then we find a nontrivial matrix representation of a subgroup of generated by some bicyclic units and their conjugations under the involution of We show that for an arbitrary finite group G our construction need not lead to a free product. At the end we shortly discuss possibility of constructing subgroups isomorphic to the free product of two free abelian groups of rank p\u2009−\u20091 for p\u2009>\u20093 in a similar way.