Let a group $G$ act on a finite set $\Omega$. Suppose that the corresponding permutation character has a regular component. Does it follow that $\Omega$ has a regular $G$-orbit?
(The converse is obviously true.)



Permutation characters and regular orbits
No, this is false in general, and the smallest counterexample is the Klein 4-group $G=C\_2\times C\_2$. It can act transitively on a set of order 2 in three different ways, with 3 different $C\_2$'s in the kernel. Call these sets $\Omega\_1, \Omega\_2, \Omega\_3$, and their permutation characters $1+\chi\_1, 1+\chi\_2, 1+\chi\_3$. Then the character of $\Omega=\Omega\_1\coprod\Omega\_2\coprod\Omega\_3$ is $3+\chi\_1+\chi\_2+\chi\_3$, which is 1+1+regular character of $G$, but $\Omega$ does not have a regular orbit.


