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

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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 b 1 and b 2 in U(ZG) such that 〈b 1, b 2〉 = 〈b 1 〉 ⋆ 〈b 2〉 ≌ Z p ⋆ Z p, a free product of two cyclic groups of prime order. Moreover, if G is nilpotent of class 2 and g has order p n, then also concrete generators for free products Z pk ⋆ Z p m 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.

Original languageEnglish
Pages (from-to)2771-2783
Number of pages13
JournalProceedings of the American Mathematical Society
Issue number7
Publication statusPublished - 2017


  • Free product
  • Group ring
  • Unit group


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