TITLE: Is there a known classification of regular multiplicity-free permutation groups?
QUESTION [6 upvotes]: The question is in the title, but let me clarify the terminology. I consider a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on a finite set $\Omega$.

$\Sigma$ is regular if it acts transitively and freely on $\Omega$, i.e., for any two $i,j\in \Omega$ there is a unique $\sigma\in\Sigma$ with $\sigma(i)=j$.
$\Sigma$ is multiplicity-free if its permutation character (the character of the linear representation of $\Sigma$ by permutation matrices) is the sum of distinct irreducible characters.

Examples are the permutation groups generated by a single cyclic permutation.

REPLY [9 votes]: These are the abelian regular permutation groups.  The permutation character in this case is the character of the regular representation and in the regular representation a character appears with multiplicity equal to the dimension of the irreducible representation.  So it can be multiplicity free iff all irreducibles are one dimensional which is equivalent to abelian.