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

Abstract

Let G G be a finite group and let p p be a prime. We continue the search for generic constructions of free products and free monoids in the unit group U ( Z G ) \mathcal {U}(\mathbb {Z}G) of the integral group ring Z G \mathbb {Z} G . For a nilpotent group G G with a non-central element g g of order p p , explicit generic constructions are given of two periodic units b 1 b_1 and b 2 b_2 in U ( Z G ) \mathcal {U}(\mathbb {Z}G) such that ⟨ b 1 , b 2 ⟩ = ⟨ b 1 ⟩ ⋆ ⟨ b 2 ⟩ ≅ Z p ⋆ Z p \langle b_1, b_2\rangle =\langle b_1\rangle \star \langle b_2 \rangle \cong \mathbb {Z}_p \star \mathbb {Z}_{p} , a free product of two cyclic groups of prime order. Moreover, if G G is nilpotent of class 2 2 and g g has order p n p^n , then also concrete generators for free products Z p k ⋆ Z p m \mathbb {Z}_{p^k} \star \mathbb {Z}_{p^m} are constructed (with 1 ≤ k , m ≤ n 1\leq k,m\leq n ). As an application, for finite nilpotent groups, we obtain earlier results of Marciniak-Sehgal and Gonçalves-Passman. Further, for an arbitrary finite group G G we give generic constructions of free monoids in U ( Z G ) \mathcal {U}(\mathbb {Z}G) that generate an infinite solvable subgroup.