# Is there a known classification of regular multiplicity-free permutation groups?

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.

• The regular action is multiplicity free iff the group is abelian – Benjamin Steinberg Mar 3 at 1:31

## 1 Answer

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.