The GFR problem and a theorem of Godsil.

A Frobenius group is a semi-direct product K ⋊ H with the property that the complement H acts fixed-point-freely on the non-trivial elements of the kernel K. A Cayley graph Γ = Cay(K,S) for the kernel K is called a graphical Frobenius representation (or GFR) of the Frobenius group G = K ⋊ H if G is the automorphism group of Γ. About five years ago, the second author of this paper conjectured (largely on the basis of a range of theoretical and computational observations made by the first author) that for any fixed finite group H, all but finitely many Frobenius groups with complement H have a GFR, except in cases where |H| is odd and the kernel K is abelian but is not an elementary 2-group. This conjecture was subsequently proved by Pablo Spiga, but it leaves an interesting problem which we call the GFR Problem, namely of determining the finitely many exceptions for a given group H. In this paper, we show what happens in a number of cases. For example, we prove that if K is isomorphic to ℤp 2 for an odd prime p, then no Frobenius group G = K ⋊ H has a GFR when H contains an element of order divisible by p + 1, while if K is isomorphic to ℤp n for some odd prime p and H acts on K as scalar multiplication, the only exceptions to having a GFR occur when n = 2 and p ∈ {3, 5, 7, 11, 13}, or (n,p) = (3,3).