A distance-transitive graph (DTG) is a graph in which for every two ordered pairs of vertices (u,v) and (u',v') such that the distance between u and v is equal to the distance between u' and v' there exists an automorphism mapping u to u' and v to v'. A semiregular element of a permutation group is a non-identity element having all cycles of equal length in its cycle decomposition. The paper gives a complete answer to a problem posed by a world-class scientist M. Guidici during a workshop in 2008 in Banff, Canada. More specifically, it is shown that every DTG admits a semiregular automorphism.
In this paper the authors consider whether a transitive solvable group contains a semiregular element. New families of groups without semiregular elements are constructed. It is also shown that if n is a positive integer such that gcd(n,phi(n)) = 1, then every solvable group of degree n contains a semiregular element, where phi is Euler's phi function. As a consequence, it is shown 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.