This question is pertaining to finite connected vertex-transitive graphs.


I recently read *"Transitive permutation groups without semiregular subgroup*" by Cameron, Giudici, Jones, Kantor, Klin, Marušič, Nowitz ([publisher link](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024610702003484); [MSN review](https://mathscinet.ams.org/mathscinet-getitem?mr=1920405)), where I found the concept of "*elusive groups*".


A permutation group $G$ acting on a set $X$ is called *elusive* if $G$ is transitive and contains no nontrivial semiregular subgroup (equivalently, no fixed-point-free element of prime order). 
I understand that transitive subgroups of elusive groups are elusive.


My question is: In light of the Polycirculant Conjecture, are the following statements is true?



> 
> Elusive groups cannot be the *full* automorphism group of any vertex-transitive graph.
> 
> 
> 


or



> 
> Elusive groups cannot be any transitive subgroup of the *full* automorphism group of any vertex-transitive graph.
> 
> 
> 


P.S. I am trying to understand how an elusive Group is a hindrance towards the Polycirculant Conjecture.


(Previously [posted on MathSE](https://math.stackexchange.com/questions/3291579/elusive-groups-and-vertex-transitive-graphs).)



Elusive groups and vertex-transitive graphs
Any elusive group of degree $n$ is a subgroup of the full automorphism group of the complete graph $K\_n$, so your second statement is not true.


The Polycirculant Conjecture asserts that the full automorphism group of a digraph contains a derangement of prime order, i.e, is not elusive. 


But while there are various families of elusive groups known, none of these groups are the full automorphism group of a digraph. 


