Relative order and spectrum in free and related groups

Abstract

We consider a natural generalization of the concept of order of an element in a group [Formula: see text]: an element [Formula: see text] is said to have order [Formula: see text] in a subgroup [Formula: see text] (respectively, in a coset [Formula: see text]) of [Formula: see text] if [Formula: see text] is the first strictly positive integer such that [Formula: see text] (respectively, [Formula: see text]). We study this notion and its algorithmic properties in the realm of free groups and some related families. Both positive and negative (algorithmic) results emerge in this setting. On the positive side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set of preorders (i.e. the set of elements of a given order) with respect to finitely generated subgroups are always computable in free and free times free-abelian groups. On the negative side, we provide examples of groups and subgroups having essentially any subset of natural numbers as relative spectrum; in particular, non-recursive and even not computably enumerable sets of natural numbers. Also, we take advantage of Mikhailova’s construction to see that the spectrum membership problem is unsolvable for direct products of nonabelian free groups.