Fixity of elusive groups and the polycirculant conjecture

2021/02/23 by Arezoomand, Majid
#05E18 #20B25 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2102.11900

Abstract

Let G≤\rm Sym(Ω) be transitive. Then G is called elusive on Ω if it has no fixed point free element of prime order. The 2-closure of G, denoted by G^(2),Ω, is the largest subgroup of \rm Sym(Ω) whose orbits on Ω×Ω are the same orbits of G. G is called 2-closed on Ω if G=G^(2),Ω. The polycirculant conjecture states that there is no 2-closed elusive group. In this paper, we study the fixity of elusive groups, where the fixity of G is the maximal number of fixed points of a non-trivial element of G. In particular, we prove that there is no 2-closed elusive solvable group of fixity at most 5, a partial answer to the polycirculant conjecture.

Related

• An infinite family of counterexamples to the Polycirculant Conjecture
• Elusive groups from non-split extensions
• Almost elusive classical groups
• Almost elusive permutation groups
• The quasiprimitive almost elusive groups