Fixed point free automorphisms of groups related to finite fields (Q439069)

This paper studies fixed-point-free (FPF) automorphisms of certain affine groups \(G\) over finite fields. (Of course, an automorphism of a group is called FPF iff it has no fixed point \textit{other than the neutral element of \(G\).}) More precisely, let \(F\) be a finite field with \(q=p^n\) elements and \(C\) the subgroup of order \(d\) of \(F^\times\) (so \(d\) must divide \(q-1\)). Then \(G\) is taken to be the semidirect product of \(F\) (additive) and \(C\) (multiplicative). The author shows among other things: If \(d\) is even, then \(G\) has no FPF automorphisms. If \(d\) is the largest odd divisor of \(q-1\), then FPF automorphisms of \(G\) exist if and only if \(p\) is a Fermat prime. For Fermat primes \(p\) and odd exponents \(p\), the FPF automorphisms are determined, and their precise count is given (it is roughly \(\phi(n)q^2\)). (If \(n=1\) and \(p\) is a Fermat prime, then we have \(d=1\) and \(G\) coincides with the additive group \(F\), which obviously admits FPF automorphisms.) The methods are direct and elementary. --- Apparently this paper is an outgrowth of the author's long-standing and ongoing work on nonclassical Hopf Galois structures on extensions of fields. If \(L/K\) is \(\Gamma\)-Galois in the classical sense, then FPF automorphisms of \(\Gamma\) give rise to nonclassical Hopf Galois structures. See the paper of \textit{L. N. Childs} and \textit{J. Corradino} [``Cayley's theorem and Hopf Galois structures for semidirect products of cyclic groups'', J. Algebra 308, No. 1, 236--251 (2007; Zbl 1119.16037)].