Fixed-point-free elements in \(p\)-groups.
It is known that every finite transitive permutation group \(G\) of degree \(>1\) has a fixed-point-free \(p\)-element for some prime \(p\) [see \textit{B. Fein, W. M. Kantor} and \textit{M. Schacher}, J. Reine Angew. Math. 328, 39-57 (1981; Zbl 0457.13004)]. It has been conjectured by \textit{P. J. Cameron} [Permutation groups. Cambridge: Cambridge University Press (1999; Zbl 0922.20003), Problem 5], that for each prime \(p\) and positive integer \(b\) there exists a bound \(f(p,b)\) such that any transitive permutation group of degree \(p^ab\) with \(a>f(p,b)\) has a fixed-point-free \(p\)-element. Cameron later offered a stronger conjecture: there is a bound \(F(m,p)\) such that if \(G\) is \(p\)-group which is a permutation group with \(m\) orbits, each of size at least \(p^{F(m,p)}\), then \(G\) contains a fixed-point-free element. The main result of the present paper is a counter-example to the second conjecture. The earlier conjecture remains open.
fixed-point free elements of prime order
scientific article; zbMATH DE number 5834350