Abstract
It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally $2$-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. It is the purpose of this paper to prove that vertex-transitive graphs of order $3p^2$, where $p$ is a prime, contain semiregular automorphisms.
Highlights
It is known that every finite transitive permutation group contains a fixed-point-free element of prime power order, but not necessarily a fixed-point-free element of prime order [3, 5]
In 1997 Klin generalized the semiregularity problem conjecturing that every transitive 2-closed permutation group contains a semiregular element – the term polycirculant conjecture is sometimes used for the semiregularity problem in this wider context. (Recall that for a finite permutation group G on a set V the 2-closure G(2) of G is the largest subgroup of the symmetric group Sym(V ) containing G and having the same orbits as G in the induced action on V × V .) The problem has spurred a lot of interest in the mathematical community producing several partial results
Giudici [9] settled the question for quasiprimitive group actions, leaving as one of the main open cases graphs admitting solvable group actions
Summary
It is known that every finite transitive permutation group contains a fixed-point-free element of prime power order (see [5, Theorem 1]), but not necessarily a fixed-point-free element of prime order (which is equivalent to existence of a semiregular element) [3, 5]. It is the object of this paper to prove the existence of semiregular automorphisms in vertex-transitive graphs of order 3p2, where p is a prime. A vertex-transitive graph of order 3p2, where p is a prime, admits a semiregular automorphism.
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