Linear recurring sequence subgroups in finite fields. (Q1414017)

Let \(f(t)=t^2-\sigma t - \rho\) be an irreducible quadratic polynomial over a finite field \(F\) of prime order \(p\) with \(\rho \not = 0\). A multiplicative subgroup \(M\) of a finite extension of \(F\) is called an \(f\)-subgroup if \(M\) is written as \(M=\{ \mu_0=1, \mu_1, \ldots, \mu_{m-1}\},\) where \(\mu_{i+2}=\sigma \mu_{i+1} + \rho \mu_{i}\) for all \(i \pmod m\). Let \(\zeta\) be a root of \(f\). Then \(M=\{ 1, \zeta, \zeta^2, \ldots , \zeta^m, \ldots \}\) is clearly an \(f\)-subgroup, and the \(f\)-subgroups \(M\) of this form is called standard. The authors study the condition that \(M\) is not standard, i.e., nonstandard. The main result of this paper is the following. Let \(M\) be an \(f\)-subgroup with the order \(| M| \) of \(M > 4\). Then \(M\) is standard if and only if both \(| M| \not = p^2 -1\) and \(| M| \) does not divide \(2(p-1)\). Some examples are also given.