Some basic results on finite linear recurring sequence subgroups

TL;DR

This paper investigates finite linear recurring sequence subgroups, classifies standard and non-standard types, and constructs infinitely many non-standard examples, advancing understanding of their algebraic structure.

Contribution

It proves that finite $f$-subgroups are generated by zeros of $f$ and constructs infinitely many non-standard $f$-subgroups, answering an open question.

Findings

Finite $f$-subgroups are generated by zeros of $f$

Improved a recent theorem of Brison and Nogueira

Constructed infinitely many non-standard $f$-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, with characteristic polynomial $f$. It is called non-standard if it can be generated in a non-cyclic way (that is, not in the order $\alpha^i, \alpha^{i+1}, \alpha^{i+2} \ldots$ for a zero $\alpha$ 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.