Relative order and spectrum of subgroups

We consider a natural generalization of the concept of order of a (torsion) element: the order of $g\in G$ relative to a subgroup $H\leq G$ is the minimal $k>0$ such that $g^k\in H$; and the spectrum of $H$ is defined as the set of orders of elements from $G$ relative to $H$. After analyzing the first general properties of these concepts, we obtain the following results: (1) every set of natural numbers closed under divisors, is realizable as the spectrum of a finitely generated subgroup $H$ of a finitely generated torsion-free group $G$; (2) $F_n\times F_n$ has undecidable spectrum membership problem: there is no algorithm to decide, given a finitely generated subgroup $H$ and a natural number $k$, whether $k$ belongs to the spectrum of $H$; and (3): in free groups $F_n$ (as well as in free-times-free-abelian groups $F_n\times Z^m$) spectrum membership is solvable, and one can give an explicit algorithmic-friendly description of the set of elements of a given order $k$ relative to a given finitely generated subgroup $H$. In this second part of the talk, arguments with automata will play a central role. (This is joint work with J. Delgado and A. Zakharov)

Date and Venue

Start Date
Online Zoom meeting
End Date


Enric Ventura

Speaker's Institution

Universitat Politècnica de Catalunya



Semigroups, Automata and Languages