Abstract
In this paper, we classify 2-closed (in Wielandt's sense) permutation groups which contain a normal regular cyclic subgroup and prove that for each such group $G$, there exists a circulant $\Gamma$ such that $\mathrm{Aut} (\Gamma)=G$.
Highlights
In 1969, Wielandt [15] introduced the concept of the 2-closure of a permutation group
Let G be a finite permutation group on a set Ω, the 2-closure G(2) of G on Ω is the largest subgroup of Sym(Ω) containing G that has the same orbits as G in the induced action on Ω × Ω, and we say G is 2-closed if G = G(2)
In order to determine 2-closed groups that contain a normal regular cyclic subgroup, we study circulant digraphs, that is Cayley digraphs of cyclic groups
Summary
In 1969, Wielandt [15] introduced the concept of the 2-closure of a permutation group. We will show that there are arc-transitive digraph representations for most 2-closed groups that contain a normal regular cyclic subgroup, see the remark after Lemma 3.12. We classify 2-closed groups G that contain a normal regular cyclic group Zn. With notation, we may suppose that G = Zn G0 Zn Aut(Zn) acting naturally on Zn. We first handle the special case that n is a prime power in Subsection 3.1 and Subsection 3.2. Remark: By [14, Lemma 2.3], a connected arc-transitive circulant Γ is both normal and of lexicographic product form if and only if Γ = Cay(Z4, {1, −1}) and Aut(Γ) = Z4 Aut(Z4) In this case the orbit 1Aut(Z4) = {1, 3} = 1 + Z2 is a coset of Z2.
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