Some basic results on finite linear recurring sequence subgroups

Abstract

An f-subgroup is a linear recurring sequence subgroup, a multiplicative subgroup of a field whose elements can be generated (without repetition) by a linear recurrence relation, where the relation has characteristic polynomial f. It is called non-standard if it can be generated in a non-cyclic way (that is, not in the order αi,αi+1,αi+2… for a zero α of f), and standard otherwise. We will show that a finite f-subgroup is necessarily generated by a subset of the zeros of f. We use this result to improve on a recent theorem of Brison and Nogueira. A old question by Brison and Nogueira asks if there exist automatically non-standard f-subgroups, f-subgroups that cannot be generated by a zero of f. We answer that question affirmatively by constructing infinitely many examples.