Abstract. Let A be an abelian group and let ι be the automorphism of A defined by ι : a → a −1 . A Cayley graph Γ = Cay(A, S) is said to have an automorphism group as small as possible if Aut(Γ) = A ⋊ ι . In this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.
It is shown that a vertex-transitive graph of valency p + 1, p a prime, admitting a transitive action of a {2, p}-group, has a non-identity semiregular automorphism. As a consequence, it is proved that a quartic vertex-transitive graph has a non-identity semiregular automorphism, thus giving a partial affirmative answer to the conjecture that all vertex-transitive graphs have such an automorphism and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69-81; P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math. 167/168 (1997) 605-615]).Let G be a permutation group on a finite set V . A non-identity element of G is semiregular if it has all orbits of equal size. It is known that each finite transitive permutation group contains a fixed-point-free element of prime power order [4, Theorem 1], but not necessarily a fixedpoint-free element of prime order, that is, a semiregular element of prime order. A permutation E-mail address: dragan.marusic@guest.arnes.si (D. Marušič).
We explicitly determine all of the transitive groups of degree p 2 , p a prime, whose Sylow p-subgroup is not isomorphic to the wreath product Zp ≀ Zp. Furthermore, we provide a general description of the transitive groups of degree p 2 whose Sylow p-subgroup is isomorphic to Zp ≀ Zp, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p 2 , explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p 2 , and find all nonnormal Cayley graphs of order p 2 .
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