## Abstract

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 language | English |
---|---|

Pages (from-to) | 2771-2783 |

Number of pages | 13 |

Journal | Proceedings of the American Mathematical Society |

Volume | 145 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2017 |

## Keywords

- Free product
- Group ring
- Unit group