# Elusive groups and vertex-transitive graphs

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; MSN review), 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.)

• I have edited it, now. – user52949 Jul 14 '19 at 14:10
• If $G$ is transitive and elusive, it can just be viewed as transitive subgroup of the automorphism group of the complete graph (on the given set). So if I understand correctly, the second question has a negative answer. – YCor Jul 14 '19 at 14:37
• Please wait more than a few hours before crossposting. – verret Jul 14 '19 at 21:01

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.
• Thank you for your explanation. But does that mean we can not have a GRR on an Elusive group, say $M_{11}$? – user52949 Jul 14 '19 at 17:02
• No, it does not, I've already explained this to you (in a private communication). You are conflating abstract groups and permutation groups. Saying "$M_{11}$ is elusive" without specifying an action is meaningless. $M_{11}$ is only elusive in one particular action of degree $12$. All its other actions are non-elusive, including its regular one. – verret Jul 14 '19 at 21:04