Let \(X\subset {\mathbb {P}}_{K}^{m}\) be a smooth irreducible projective algebraic variety of dimension \(d \ge 1\) , defined over an algebraically closed field K of characteristic \(p \ge 0\) . Let \(n \ge d+1\) and \(k \ge 2\) be integers. If \(p>0\) , then we also assume k to be relatively prime to p and that \(k-1\) is not a power of p. We say that X is a generalized Fermat variety of type (d; k, n) if there is a Galois branched covering \(\pi :X\rightarrow {\mathbb {P}}_{K}^{d}\) , with a group of Deck transformations \({\mathbb {Z}}_k^n\cong H<\textrm{Aut}(X)\) , whose branch divisor consists of \(n+1\) hyperplanes in general position (each one of branch order k). In this case, the group H is called a generalized Fermat group of type (d; k, n). We prove that, if either (i) \(p=2\) or (ii) \(p \ne 2\) and \((d;k,n) \notin \{(2;2,5), (2;4,3)\}\) , then a generalized Fermat variety of type (d; k, n) has a unique generalized Fermat group of that type.
