Fixed point free automorphisms of groups related to finite fields

Let G = Fqo 〈β〉 be the semidirect product of the additive group of the field of q = p elements and the cyclic group of order d generated by the invertible linear transformation β defined by multiplication by a power of a primitive root of Fq. We study endomorphisms of G. We find an arithmetic condition on d so that every endomorphism is determined by its values on (1, 1) and (0, β). When that is the case, we determine the fixed point free endomorphisms that are abelian (i.e. factor through an abelian quotient of G) and the fixed point free automorphisms of G. If d equals the odd part of q − 1 then we count the fixed point free automorphisms of G–such exist if and only if p is a Fermat prime. Introduction Let p be an odd prime, q = p, Fp the field of p elements, A = Fq the field of q = p elements. Then under addition, A = Fp is an elementary abelian p-group of rank n. Let x be a primitive root of Fq. Let β be the automorphism of A given by β(f) = xf , multiplication by x, for all f in A, where b 6≡ 0 (mod q−1). Then β generates a cyclic subgroup of GLn(Fp) of order d where d = (q − 1)/(q − 1, b). Let G = Ao 〈β〉, the semidirect product of A and the cyclic group generated by β, where β acts on A as above. We will denote elements of G by (f, β). Then the operation is (f, β)(g, β) = (f + xg, β). One sees easily that G has trivial center. We are interested in determining the fixed point free endomorphisms of G. In addition to the intrinsic interest in finding endomorphisms of G, such endomorphisms are of interest in Galois theory. In 1968, Chase and Sweedler [CS68] introduced the notion of Hopf Galois extension, and in 1987 Greither and Pareigis [GP87] observed that a given Galois Date: November 25, 2009. My thanks to the Mathematics Department at Virginia Commonwealth University for its hospitality while this research was conducted. 1