Free products in the unit group of the integral group ring of a finite group

Janssens, Geoffrey [UCL] Jespers, E. Temmerman, D.

Let G be a finite group and let p be a prime. We continue the search for generic constructions of free products and free monoids in the unit group U(ZG) of the integral group ring ZG. For a nilpotent group G with a non-central element g of order p, explicit generic constructions are given of two periodic units b1 and b2 in U(ZG) such that 〈b1, b2〉 = 〈b1 〉 ⋆ 〈b2〉 ≌ Zp ⋆ Zp, a free product of two cyclic groups of prime order. Moreover, if G is nilpotent of class 2 and g has order pn, then also concrete generators for free products Zpk ⋆ Zpm are constructed (with 1 ≤ k, m ≤ n). As an application, for finite nilpotent groups, we obtain earlier results of Marciniak-Sehgal and GonçalvesPassman. Further, for an arbitrary finite group G we give generic constructions of free monoids in U(ZG) that generate an infinite solvable subgroup. © 2017 American Mathematical Society.