Frobenius group

A permutation group $G$ on a set $X$ is Frobenius if no non-trivial element of $G$ fixes more thanone element of $X$. Generally, one also makes the restriction that at least one non-trivial elementfix a point. In this case the Frobenius group is called non-regular.

The stabilizer of any point in $X$ is called a Frobenius complement, and has the remarkableproperty that it is distinct from any conjugate by an element not in the subgroup. Conversely,if any finite group $G$ has such a subgroup, then the action on cosets of that subgroup makes$G$ into a Frobenius group.