On Semiregular Elements of Solvable Groups

Abstract

We consider whether a transitive solvable group contains a semiregular element (a fixed point free element whose orbits are all of the same length). We first construct new families of groups without semiregular elements. We also show that if n is a positive integer such that gcd(n, ϕ(n)) = 1, then every solvable group of degree n contains a semiregular element, where ϕ is Euler's phi function. As a consequence, we show that for such n if every quasiprimitive group of composite degree m dividing n is either A _m or S _m , then every transitive group of degree n contains a semiregular element.

Key Words:

• Cayley number
• Elusive group
• Polycirculant conjecture
• Semiregular element
• Solvable group

2000 Mathematics Subject Classification:

• 20B05
• 20B25

ACKNOWLEDGMENT

The second author was supported in part by “Agencija za raziskovalno dejavnost Republike Slovenije,” research program P1-0285.

Notes

Communicated by A. Turull.