A note on transitive permutation groups of degree twice a prime

In this note, we consider transitive permutation groups of degree 2p, where p is an odd prime, that admit blocks of imprimitivity of size 2 but no blocks of imprimitivity of size p. Primitive permutation groups of degree twice a prime were first considered by Wielandt [6], while more recently, imprimitive permutation groups of degree twice a prime were studied by Lefèvre [2], Marušič [3], and Marušič and Potočnik [4]. In this note we present a new result on imprimitive permutation groups of degree twice a prime that serves as a crucial tool in the classification of homogeneously almost self-complementary graphs of order four times a prime in [5]. lem:GeFolk Theorem 1 Let p be an odd prime, V a set of size 2p, G a transitive permutation group on V , and P a Sylow p-subgroup of G. Then P has two orbits on V , each of size p. Suppose further that the orbits of P are not blocks of imprimitivity for G, but that there exists a G-invariant partition B of V into blocks of size 2. Let K = Ker(G → GB) denote the kernel of the induced action of G on B. Then one of the following occurs: (i) |K| ≤ 2, GB is a non-solvable 2-transitive group, and for every B ∈ B and v ∈ B the stabilizer Gv acts transitively on the set V \B; (ii) |K| ≥ 4 and either ∗This author gratefully acknowledges support by the Natural Sciences and Engineering Research Council