  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.
