Constructing free products of cyclic subgroups inside the group of units of integral group rings

Abstract

It has been proved in Janssens, Jespers, and Temmerman [Proc. Amer. Math. Soc. 145 (2017), pp. 2771–2783] that if h h is an element of prime order p p in a finite nilpotent group G G and u = h + ( h − 1 ) g h ^ ∈ Z G u=h+(h-1)g\widehat {h}\in \mathbb {Z}G , u ∉ G u\not \in G , then ⟨ u ∗ , u ⟩ ≈ C p ∗ C p \langle u^*,u\rangle \approx C_p\ast C_p . We offer a simple geometric approach to generalize this result.