Regular sections groups (Q1904525)

Let \(K= GF(q)\) be a finite field of characteristic \(p\), and let \(L= GF (q^s)\). A regular sections group is a subgroup \(G\) of the multiplicative group \(L^*\) such that the different values of \(|H\cap G|\), \(H\) a \(K\)-hyperplane of \(L\), appear with the same multiplicity. The author proves that this situation arises if and only if \((q^s -1) \text{ gcd} (q-1, n) n^{-1} (q-1)^{-1}\) divides \(p-1\), where \(n\) is the order of \(G\).