We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Relative order and spectrum in free and related groups.
- Authors
Delgado, Jordi; Ventura, Enric; Zakharov, Alexander
- Abstract
We consider a natural generalization of the concept of order of an element in a group G : an element g ∈ G is said to have order k in a subgroup H (respectively, in a coset H u) of G if k is the first strictly positive integer such that g k ∈ H (respectively, g k ∈ H u). 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.
- Subjects
NATURAL numbers; ORDERED groups; NONABELIAN groups; SYLOW subgroups; FREE groups
- Publication
Communications in Contemporary Mathematics, 2024, Vol 26, Issue 1, p1
- ISSN
0219-1997
- Publication type
Article
- DOI
10.1142/S0219199722500663