Fixed-Point-Free p -Elements in Transitive Permutation Groups

Abstract

We give sufficient conditions for a finite permutation group to contain a fixed-point-free permutation of/>-power order for a given prime p.THEOREM 2. Let p be an odd prime number and a ^ 1. Ifp + 1 < b < §(/> +1), then every transitive permutation group of degree p a b contains a fixed-point-free p-element, that is,p a -bistf v .ACKNOWLEDGEMENT. The author is grateful to Peter P. Palfy for helpful discussions.

Fixed-point-free elements in p-groups

“…The reader can check in [1] for some similar results on this general problem. We remark that Conjecture A (and also the case p = 2 implicitly due to Isbell) is widely open.…”

Unisingular representations of finite symplectic groups

“…Therefore, G has 11 irreducible Brauer characters. The character ρ 4 has C 3 3 in its kernel, so this can be viewed as a character of A 4 ; this is reducible modulo 2 as A 4 has a non-trivial normal 2subgroup.…”

“…Lemma 6.10. Let G = 3 D 4 (3). Then G has a unisingular absolutely irreducible representation into Sp 218 (2).…”

“…For every fixed prime p determine the set P p of integers n > 1 such that every transitive permutation group of degree n contains a fixed point free p-element. Some contribution to this problem is obtained in [3]. In particular, it is shown that p a (p + 1) ∈ P p for a > 1.…”