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

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