Relative order and spectrum in free and related groups

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