On semiregular elements of solvable groups
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 Am or Sm, then every transitive group of degree n contains a semiregular element.