Linear recurring sequence subgroups in finite fields

Given a finite field and a linear recurrence relation over it is possible to find, in a fairly “obvious” way, a finite extension of and a subgroup M of the multiplicative group of such that the elements of M may be written, without repetition, so as to form a cyclically closed sequence which obeys the recurrence. Here we investigate this phenomenon for second-order recurrences; the situation in which has prime order and the characteristic polynomial of the relation is irreducible over is described.