Second order linear sequence subgroups in finite fields (Q2426458)

Let \(\mathbb F\) be a finite field and \(f(t)=t^2-\sigma t-\rho\) a quadratic with coefficients in \(\mathbb F\). An \(f\)-sequence is a sequence satisfying the recurrence relation \(\mu_{i+2}=\sigma\mu_{i+1}+\rho\mu_i\) for all integers \(i\). If \(f(t)\) has roots \(g, h\), then the subgroups \(\langle g\rangle, \langle h\rangle\) are cyclic \(f\)-sequences. The multiplicative subgroup \(M = \{\mu_0,\ldots,\mu_{| M| -1}\}\) in \(K^*\), where \(K\) is some finite extension of \(\mathbb F\), is called an \(f\)-subgroup if its elements in an appropriate order fit into an \(f\)-sequence \(\{\ldots,\mu_0=1, \mu_1,\ldots,\mu_{| M| -1},\ldots\}\). An \(f\)-subgroup is called non-standard if there is such a representation with \(f(\mu_1)\neq0\); otherwise it is standard. The terminology is explained by the observation that an \(f\)-subgroup is standard if and only if the only \(f\)-sequences which represent \(M\) in this way are cyclic. The authors propose a method for finding non-standard subgroups and give several examples. Two of their results are: (1) Suppose \(q=p^n\) where \(p\) is a prime, \(i| n, n/i\) is odd and \(e| (p^n-1)/(p^i-1)\). Then the subgroup of order \(e(p^{2i}-1)\) in \(\mathbb F_{q^2}^*\) is non-standard. (2) Let \(q=p^2\) where \(p\) is an odd prime, and suppose \(f(t)\) is irreducible and \(M\) is an \(f\)-subgroup of \(\mathbb F_{q^2}^*\) with \(| M| >4\). Then \(M\) is standard if and only if both \(| M| \neq q^2-1\) and \(| M| \nmid 2(q-1)\). Suppose \(M\) is non-standard. The aim is to exhibit \(M\) or to reach a contradiction and deduce that \(M\) is standard. The method involves finding a polynomial, \(p(t)\) say, which permutes the elements of \(M\) and observing that the constant terms of \(p(t)^s \bmod t^{| M| }-1\) is 0 if \(s\not\equiv 0\bmod | M| \) and 1 if \(s\equiv 0\bmod | M| \). Under the assumptions of (2), \(M\) can be represented by \(\{\alpha g^i+\beta h^i\}\) and \(p(t)=\alpha t+\beta t^q\) permutes the elements of \(M\). The constant term of \(p(t)^s\) for suitably chosen values of \(s\) is then found with considerable ingenuity by using Lucas' theorem and other identities involving binomial coefficients.