On the Unique Order Property and Subgroup Structure of Roots of Unity in Finite Groups

Abstract

This paper investigates the structural properties of the set of [Formula: see text]th roots of unity in a finite group [Formula: see text], defined as [Formula: see text]. We focus on two primary inquiries: the conditions under which these sets constitute subgroups and the validity of the unique order property, which asserts that [Formula: see text] implies [Formula: see text]. Regarding the former, we prove that for [Formula: see text], the set [Formula: see text] forms a (normal) subgroup of [Formula: see text] if and only if [Formula: see text]. This result parallels the Frobenius conjecture, which was settled in the 1990s. Regarding the latter, we demonstrate that the unique order property holds for all [Formula: see text]-groups, as well as for any indices [Formula: see text] where [Formula: see text] and [Formula: see text] are subgroups. Furthermore, we characterize groups in which every set of roots of unity is a subgroup, establishing that this occurs if and only if [Formula: see text] is nilpotent and [Formula: see text] is a subgroup for every [Formula: see text]-Sylow subgroup [Formula: see text] and [Formula: see text]. Under these conditions, [Formula: see text] contains exactly [Formula: see text] distinct root of unity subgroups. Finally, we present the smallest counterexample to the unique order property - a group of order 96, and pose the open problem of its universal characterization.